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The adaptive adjustment of digital data receivers using pre-detection filter This item was submitted to Loughborough University's Institutional Repository by the/an author.

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A Doctoral Thesis.

Submitted in partial fulfilment of the requirements

for the award of Doctor of Philosophy at Loughborough University.

Metadata Record: https://dspace.lboro.ac.uk/2134/27569 Publisher:

c

S.Y. Ameen

Rights: This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 2.5 Generic (CC BY-NCND 2.5) licence. Full details of this licence are available at: http://creativecommons.org/licenses/bync-nd/2.5/

Please cite the published version.

This item was submitted to Loughborough University as a PhD thesis by the author and is made available in the Institutional Repository (https://dspace.lboro.ac.uk/) under the following Creative Commons Licence conditions.

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LOUGHBOROUGH UNIVERSITY OF TECHNOLOGY LIBRARY AUTHOR/FILING TITLE

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009 4205 02

THE ADAPTIVE ADJUSTMENT OF DIGITAL DATA RECEIVERS USING PRE-DETECTION FILTER

BY SIDDEEQ YOUSIF AMEEN

A Doctoral Thesis submitted in partial fulfilment of the requirements of the award of the degree ofDoctor of Philosophy of the Loughborough University of Technology

May 1990

Supervisor: Dr. S. C. Bateman Department of Electronic and Electrical Engineering

,,

@by S. Y. AMEEN

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ACKNOWLEDGMENTS I would like to express my thanks to my supervisor Dr. S. C. Bateman for his mvaluable help, advice and encouragement throughout this project without which this work could not have been completed. My director of research, Professor A. P. Clark, was the architect of this research project. Always at hand for advice, conversation and cntical evaluauon of theories and results. For all this and much more, I am deeply grateful and very sorry for his death. The financial support of IRAQI government which made this work poSSible is gratefully acknowledged. On personal level I would hke to thank my family for therr support and encouragement. Most important of all Mona, my wife, who has shown wonderful patience, help, encouragement and understandmg throughout the course of this work.

1

ABSTRACT This thesis is concerned with the adaptive adjustment of digital data receivers employed m synchronous senal data transmission systems that use quadrature amplitude modulation The receiver employs a pre-detection filter that forms the first part of a decision feedback equalizer or else is used ahead of a near maximum-likelihood detector. The filter is Ideally adjusted such that the sampled Impulse response of the channel and filter is mimmum phase. In earlier versions of the filter, when used as part of a decision feedback equalizer it was adjusted by means of the gradient (LMS) algorithm to mimmize the mean-square error m the signal at the detector input. Alternauvely, a more recent technique has enabled the filter to be adJusted directly and in a relatively simple manner from the estimate of the sampled impulse response of the linear baseband channel, which can be determined both accurately and rapidly. The relauve performances and merits of both earlier and recent techniques are investigated for different telephone channels and at transmission rates of9600 and 19200 bit/s. The results confirm that both techniques achieve nearly the same performance at the higher signal-to-noise ratios. The thesis then considers several developments of the more recent technique All of these require a knowledge of the roots of the z-transform of the sampled impulse response of the channel that he outside the unit circle m the z-plane. Several novel algonthms to identify and locate the given roots have been mvestigated Fmally, the technique has been developed for use over HF radio links. Several modified verswns of the original algorithm previously investigated for telephone channels, have also been developed and studied.

11

LIST OF CONTENTS

~

1

LIST OF PRINCIPAL SYMBOLS

vii

INTRODUCTION

1

1.1

BACKGROUND

1

1.2

OUTLINE OF INVESTIGATION

4

2

DATA

TRANSMISSION

SYSTEMS

OVER

THE

7

TELEPHONE NETWORK 2.1

INTRODUCTION

2.2

GENERAL MODEL OF THE DATA TRANSMISSION

7

SYSTEM 2.3

7

ASSESSMENT OF THE DISTORTION PRESENT IN THE SAMPLED IMPULSE RESPONSE OF A BASEBAND

~

CHANNEL

10

2.4

TELEPHONE CIRCUITS

11

2.5

ATTENUATION AND GROUP DELAY DISTORTION

14

2.6

NOISE

15

2.7

MODEL OF THE DATA TRANSMISSION SYSTEM USING QAM SIGNAL

2.8

-

SAMPLED IMPULSE RESPONSES OF CHANNELS USED IN COMPUTER SIMULATION$

~

3

-

CHANNEL EQUALIZATION

3.1 " 3.2

3.3

17

23

40

INTRODUCTION

40

LINEAR EQUALIZER

40

NONLINEAR EQUALIZER

44

lll

34

'

4

DECISION FEEDBACK EQUALIZER

46

3.4.1

Zero Forcing (ZF) Equalizer

48

3.42

Minimum Mean-Square Error (MMSE) Equalizer

51

3.4.3

Comparision of Equalizers

54

a

In the presence of phase distortion

54

b

In the presence of amplitude distortion

57

ADJUSTMENT OF THE PRE-DETECTION FILTER

63

4.1

INTRODUCTION

63

4.2

ADJUSTMENT SCHEME 1

65

43

ADJUSTMENT SCHEME 2

70

4.4

ADJUSTMENT SCHEME 3

73

4.5

ADJUSTMENT SCHEME 4

74

4.6

ASSESSEMENT OF RESULTS AND DISCUSSION

76

5

ALGORITHMS FOR Tf-JE ADJUSTMENT OF THE PRE-DETECTION FILTER

96

5.1

INTRODUCTION

96

5.2

ROOT-IDENTIFICATION ALGORITHMS

98

5.2.1

Schur Algorithm

98

5.2.2

Nyquist Criterion

101

5.3

ROOT-FINDING ALGORITHMS

105

5.3 1

Algorithm 1

105

5.3.2

Algorithm2

111

5.3.3

Algonthm 3

113

5.3.4

Algonthm4

117

5.3.5

Algorithm 5

118

IV

5.4

' 6

ASSESSMENT OF RESULTS AND DISCUSSION

ADAPTIVE DECISION FEEDBACK EQUALIZERS

121 159

6.1

INTRODUCTION

159

6.2

FEEDFORWARD TRANSVERSAL FILTER CHANNEL ESTIMATOR

159

ADAPTIVE DECISION FEEDBACK EQUALIZERS

163

'63 63 1

Equalizer 1

163

6.3.2

Equalizer 2

167

6.3.3

Equalizer 3

170

64 7

ASSESSMENT OF RESULTS AND DISCUSSION

ADAPTIVE ADJUSTMENT OF THE RECEIVER OVER HF RADIO LINKS

172

189

7.1

INTRODUCTION

189

7.2

MODEL OF SYSTEM

189

7.3

ADJUSTMENT ALGORITHMS

194

7.4

7.5

7.3.1

Adjustment algorithm 1

196

7.3.2

Adjustment algorithm 2

197

7.3.3

Adjustment algorithm 3·

198

7.3.4

Adjustment algorithm 4

199

7.3.5

Adjustment algorithm 5

201

DECISION FEEDBACK EQUALIZERS OPERATING OVER HF CHANNELS

204

ASSESSMENT OF THE RESULTS AND DISCUSSION

207

V

......

8

'

CONCLUSION

230

8.1

COMMENT ON THE RESEARCH

230

8.2

SUGGESTION FOR FURTHER INVESTIGATION

232

REFERENCES

234

APPENDICIES LIST OF COMPUTER SIMULATION PROGRAMS

242

APPENDIX 1 CALCULATION OF THE SAMPLED IMPULSE RESPONSE

OF

THE

LINEAR

BASEBAND

'

CHANNEL OVER TELEPHONE NETWORKS

242

APPENDIX 2 LINEAR PRE-DETECTION FILTER ADJUSTMENT SCHEME 1

249

APPENDIX 3 LINEAR PRE-DETECTION FILTER ADJUSTMENT SCHEME2 APPENDIX 4

253

LINEAR PRE-DETECTION FILTER ADJUSTMENT SCHEME4

APPENDIX 5

THE

256

SCHUR

ROOT-IDENTIFICATION

ALGORITHM APPENDIX 6 THE

APPENDIX 7

265

NYQUIST

ROOT-IDENTIFICATION

ALGORITHM

268

ROOT-FINDING ALGORITHM 4

270

APPENDIX 8 DECISION FEEDBACK EQUALIZER 1

277

APPENDIX 9 ADJUSTMENT ALGORITHM 4 OVER HF RADIO LINKS

288

APPENDIX 10 DECISION FEEDBACK EQUALIZER H5 OVER HF RADIO LINKS

297

VI

LIST OF PRINCIPAL SYMBOLS a(t) & A(f)

Impulse response and transfer function of a filter. Attenuation of telephone circmt and eqmpment filter at frequency f. z-transform of one-tap feedback transversal filter in Fig. 5.22.

B

(n+g+ I)-component row vector given by Eqn. 3.45. z-transform of two-tap feedforward filter in Fig 5.40.

b(t) & B(f)

Impulse response and transfer function of b filter.

c

Positive constant.

c

Equalizer tap gains in Sections 3.1 and 6 3.3.

c(t) & C(f)

Impulse response and transfer function of bandpass filter C.

d

Threshold level for determmmg convergence of iterative process.

D(z)

z-transform of hnear pre-detecnon filter.

D'(z)

z-transform of reversed order sequence corresponding to D(z).

E[x]

Expected value of x.

E(z)

z-transform of the combined sampled impulse response of the channel and pre-detection filter.

e,

Error signal at t=iT. Carrier frequency m QAM system in Hertz.

/(z)

First denvative of f(z).

((z)

Second derivative of f(z).

g+l

Number of components in a sampled impulse response of the hnear baseband channel.

g(t) & G(f)

Impulse response and transfer function of the filter G.

gd,(f)&gd,(f)

Group delay of telephone circuit and equipment filter.

gd

Average group delay.

Vll

h(t) & H(f)

Impulse response and transfer function of the telephone circuit.

CL(j), HL(j) & GL(j) Low frequency components of C(f), H(f) & G(f).

n+l

Number of taps of the lmear pre-detectiOn filter.

n(t)

White Gaussian noise waveform at the input of telephone circuit.

r(t)

Received waveform.

{r,}

Sequence of received samples that is obtruned by samplmg r(t) at t=iT, i=l, 2, ..... .

R,(f)

Transfer function of the receiver filter set at frequency f given by Eqn. 2.20.

f,

Estimate of r,.

s,

The 1m transmitted data symbol.

s,

The detected value of s,.

Sa,l'

sb,•

The real and the imaginary part of s,

s,

(n+g+ I)-component row vector given by Eqn. 3 37.

T,(f)

Transfer function of the transmitter filter set given by Eqn. 2 19.

Um(z)

Deflated form of the polynorrual Y(z) after removal of m roots.

{u.}

Noise samples at the pre-detection filter output.

v,

Output signal from the linear pre-detection filter.

w(t)

A Gaussian random process with zero mean.

{w,}

A sequence obtained by sampling w(t) at t=iT, i=l, 2, ....

x(t)

QAM signal given by Eqn. 2.6.

y

Sampled impulse response of the linear base band channel.

Y(z)

z-transform of the sampled Impulse response of the lmear base band channel.

y

Estimate of Y.

Yr(z)

Truncated form of Y(z).

Vlll

Factor of Y(z) with all its roots inside the unit circle. Factor of Y(z) with all its roots outside the unit circle.

z.

(n+ 1)-upper triangular square matnx given by Eqns. 3.46 and 4.38.

o(r)

Dirac function (unl!Impulse). Variance of u,.

8

posotlve constant. Mean-square value of the real or Imaginary part of s,. Variance of real or imaginary part of w,. Parameter given in Eqns. 3.49, 4.21 and 7.24. Mean-square error due to additive nmse only given by Eqn. 4 20. Parameter given in Eqn. 6.12. Parameter giVen in Eqns. 4.18 and 6.1 0. Parameter given in Eqns. 4.19 and 6 11. Parameter given m Eqns. 7.18.

e.

Parameter given in Eqn. 5.22. Negative of the roots of Y(z).

~)

Negative of the reciprocal of a root of Y(z) that he outside the umt crrcle.

p(t)

Stationary zero-mean real-valued Gaussian random process. Estimate of the negative of the reciprocal of the m'h roots at the 1terat10n.

i''

Estimate for the biggest roots of Yr(z) Threshold level for determimng convergence of iterative process. Estimate of ~•.

ix

Phase angle. Channel estimator step s1ze (Eqn. 6.3). Variation around the contour C. Sampling penod in frequency domam. Gradient algonthm step size.

e

Variation of around the unit circle Transmitted data. Detected value of Xk·

*

ConvolutiOn operator.

super script *

Complex conjugate.

super script T

Transpose.

X

CHAPTER!

INTRODUCTION

1.1 BACKGROUND

The widespread use and the increasing availabllity of low-cost digital data transmission systems has created an mcreasing demand for more efficient methods of transmissiOn. Furthermore, the rapid advances in digital computers and silicon technology have increased the reqmrement fortransmittmg data signals at the highest rates over a wide range of communication channels This in turn has made the problem of digital communication through hnear channels that exhibit both amplitude and phase distortion important and practical. Different media are used for digital data transmission, but the most Important of these due to therr very widespread existence are voice-frequency channels over the telephone network and HF radio links [1-10]. The public switched telephone network (PSTN) was used pnmanly to carry analogue speech signals only [3-5,8]. It followed that the spectral characterisncs of such channels were optimized, over time, to achieve a good noise performance and an acceptable quality of service, the latter being a highly subjective measure [3,5]. However, the use of these voice-frequency channels for transmission of greater volumes ofhigher-speed data traffic has shown that very specialist and sophisticated methods of signal processing must be employed at the receiver m order to achieve an acceptable error rate performance, which IS, itself, a highly definable measure [3,9,10] In essence, amplitude and phase distortion present m a data signal will cause It to spread out in time, eventually resulting in intersymbol interference, where adJacent data elements overlap in time [3,10-18]. It is m this sense (1 e taking into account the distortion and the recognition that different channels will introduce different amounts of distortiOn), that modern, high-speed digital data receivers must be adaptive m structure [3,13-18]

1



The high-frequency (HF) (nominally 3-30 MHz) portion of the spectrum has been of great interest for long distance radio commumcations in a variety of nulitary and CIVIlian applicatiOns for many years [3,5,7,14,17-24]. These have different properties from telephone circuits and they are normally used for point-to-pomt communications, or for isolated communication networks, and not often used as a part of the general telephone network [3,5,17]. However, HF radio lmks are subject to ionosphenc multipath propagation and fading, which makes the problem of communication over these channels difficult, even at moderate rates [5,7] The major limitations for high data rate HF transmissiOn are a result of the non-Ideal characteristics of the medmm, such as lmear distortion, rapid channel variations and severe fadmg as well as bandwidth constraints [3,5,7,25]. Therefore, m high speed digital commumcatwns, the efficient use of the available bandwidth IS often linuted by the presence of intersymbol interference caused by the non-Ideal channel charactenstics. Clearly, high speed data transmission can be severely degraded by the presence of intersymbol mterference unless the receiver employs complex equalization techmques and multi-level signallmg [3, 11, 14, 16,27] The lattens achieved by using an efficient modulation technique such as quadrature amplitude modulation (QAM), which has been used since the late 1960's to achieve reliable transmission of data at rates of upto 9600 bit/s and more [26]. In practice, and due to their easy implementatiOn using cheap, medmm-technology components, linear equalizers are often used [27-29]. For channels introducing a severe degree of bandlimiting, coupled with the requrrement of a high transmission rate, a sigmficantly better performance can be achieved under comparable operation conditions, by the use of a pure nonlinear equalizer [27 -30]. The latter uses a method of decisiOn directed cancellation of mtersymbol interference and can suffer from error extension effects [27]. An alternative method of equalization is to use the deCision feedback equalizer. This type of equalizer mcludes an adaptive lmear pre-detection filter placed ahead of the nonlinear equalizer structure itself. The decision feedback equalizer often achieves a better tolerance to nOise than a lmear and a pure nonlinear equalizer [3,27]. However, computer simulations studies have shown that partial equalization of the channel, along With the use of a fmrly sophisticated detection scheme such as near maximum-likelihood detection may

2

well result in a system which is cost effective (in terms of computational effort) and also gives a supenor performance when compared with classical equahzanon techniques [ 11,17 ,31]. The decision feedback equalizer and the partial equalization that is employed by the near maximum-likelihood detector both use a lmear transversal filter called the pre-detection filter. In the earher versiOns of the filter, it was adJusted by means of the gradtent (least mean-square error) algorithm or equivalent techniques to mmimize the mean-square error m the output stgnal at the detector mput [3,27,29,31-33]. Unfortunately, when the channel introduces severe amplitude dtstortion, correct convergence is not necessanly achieved by this arrangement, and the rate of convergence may be very slow [3,32]. Altemauvely, undue complexity may be involved in the system [17] However, a recent development has enabled the filter to be adjusted drrectly and in a relatively simple manner from the esumate of the sampled impulse response of the channel, which can be determined both accurately and rapidly even in the presence of severe amplitude distortion [3,17,33-38]. Thts has opened the way to widespread research. It is essentially a root-findmg algonthm that determines, m sequence, the roots (zeros) of the z-transform of the sampled impulse response of the linear baseband channel, that lie outside the umt ctrcle in the z-plane It then uses the knowledge of these roots to determine the tap gains of the linear pre-detection ftlter and to form an estimate of the sampled impulse response of the channel and filter. The pre-detection filter at all times introduces an orthogonal transformanon into the received signal, bemg constramed to be an allpass network [17,27 ,39]. When the filtens correctly adjusted, the zeros of the z-transform of the sampled impulse response of the channel and filter are denved from those of the z-transform of the sampled impulse response of the channel by replacmg all zeros of the latter that he outside the umt circle by the complex conjugate of thetr rectprocals, all remaimng zeros bemg left unchanged. The mm of the work presented m this thesis is to develop satisfactory methods that can be used m the adJUStment ofthe pre-detection ftlter for use over telephone circuits and HF radiO links which are being used for transmission of data at high speeds (9600 bit/r and above over voice-frequency channels). The study also aims to compare the performance of the developed methods with more conventional soluuons.

3

1.2 OUTLINE OF INVESTIGATION

Essentially, the investigation is concerned with the senal transmission of data at 9600 bJt/s over telephone crrcmts and HF radio links. Over telephone crrcuits, a feas1billty study has been earned out in this work, looking mto the poSSibility of transmission at rates close to the Shannon limit, i.e. 19200 bit/s [40]. The main concern has been the development of methods for the adJustment of the pre-detectwn filter that can be employed by the decision feedback equal1zer or near maximum-likelihood detector. The research has been earned out by computer simulation of the modem. In th1s situatiOn, computer simulatiOn IS a valid means of evaluating the system performance because the modem can be considered as a digllal s1gnal processor performing computer like operations on set of numbers Chapter 2 provides a detailed description of the general model of a synchronous senal data transmission system. The nature of telephone Circuits and the types of signal distortion and other impairments introduced by these circuits along w1th the types and effects of noise introduced are also presented and d1scussed. The suitability of quadrature amplitude modulation (QAM) is discussed and the system model using 16-level and 64-level QAM signal constellations are also d1scussed. The model of the system using QAM signals is then presented together wJth the relauonsh1ps between the bandpass processes and signals, wllh their equivalent baseband representauon. Fmally, the sampled impulse responses of the equivalent lmear base band channels when the data symbols transmllted at 2400 and 3200 baud over d1fferent telephone circuits are calculated and recorded. Chapter 3 contains the descriptions of the linear, pure nonlinear and decision feedback equalizers These equalizers employ linear feedforward transversal filters and can be used as techniques for processing distorted received s1gnals The decisiOn feedback equalizer employing the pre-detection f1lter achieves better performances than both linear and pure nonlinear equalizers and can be adJusted eJther to minimize the mean-square error in the equalized signal or max1mize the signal-to-nmse ratio at the detector input. The latter is achieved subject to the exact equalizauon of the channel. Theoreucal comparison between the two type of decision feedback equalizers under d1fferent circumstances are then presented.

4

Chapter4 investigates several schemes that adaptively adJUSt the pre-detection filter. The filter is adJusted using two criteria, the first maximizes the signal-to-noise ratio at the input of the detector subject to exact equalization of the channel and the second mmimizes the mean-square error in the signal at the detector input. Three adjustment schemes are presented which adjust the filter according to the former cntena and a fourth scheme adjusts the filter according to the minimum mean-square error cnteria. The study aims to show the relationship between both cnteria and the complexity of the firSt three schemes. Computer Simulation studies when operating over different telephone channels are presented for various adjustment schemes, and their performances, in terms of equalization accuracy and complexity, are investigated. Chamer 5 descnbes different algorithms that have been used in the adjustment mechanism of the pre-detection filter to locate or identify all or some of the roots of the z-transform of the sampled impulse response of the linear baseband channel. These algorithms operate directly on an estimate of the raw sampled Impulse response of the channel and can, by design, be made to operate outside the normal flow of processing steps reqmred in the receiver of a data modem. The estunate of the raw sampled Impulse response is taken to be ideal estimate. For the requirement of the digital receiver to the estimate of the intersymbol interference, some of these algorithms have an extra stage, where the estimate of the sampled Impulse response of the channel and pre-detection filter is calculated. Results of computer simulation tests are then presented showing the convergence rate and the accuracy of the iterative processes. The number of numerical operations involved in the execution of each algorithm are also shown. The latter measure IS extremely Important

when

considering the development of the modem using practical digital signal processmg devices. Chapter 6 first discusses the adaptive estimation of the sampled impulse response of the linear baseband channel. It then extends the study to mclude the adaptive adJustment of the decision feedback equalizer. The decisiOn feedback equahzer here is adjusted to maximize the signal-to-noise ratio at the detector mput subject to the exact equahzation of the channel. Results of computer simulation tests over different models of telephone channels are then presented showing the channel estimator performance. The effect, of the channel estimator together with the finite number of taps in the pre-detection filter, on the equalizer performance were also

5

--------------------------------------------------------

--

--

-

invesngated. Fmally, the tolerance of the equahzerto additive white Gaussian noise was computed and compared with the Ideal case and with the cases where the equalizer was adjusted by the gradtent (LMS) or the Kalman (RLS) algorithms. In the ideal case, perfect estimanon of the sampled impulse response of the channel, theoretical root-finding algorithm and mfmite number of taps in the pre-detection filter were assumed. Chapter 7 first presents the model of the data transmission over HF rad10 lmks . Several modified verswns of the origmal algonthm previously invesngated for telephone channels have been developed and studied for use over HF radiO lmks. Results of computer simulation tests showmg the accuracy and the complexity of the algorithms over HF radiO links, that introduce vanous levels of distortion, are calculated and presented. Finally, the effects of these algorithms on the tolerance of the decision feedback equalizer to additive white Gaussian noise, when the algorithms are used to adJUSt the equalizer, were also investigated for different HF radio links

6

CHAPTER2

DATA TRANSMISSION SYSTEMS OVER THE TELEPHONE NETWORK

2.1 INTRODUCTION

In the last three decades the demand for efficient and high speed data transmissiOn systems has mcreased. Comprehensive studies to put mathematical models of reliable data transmission systems have been based on Shannon's result on the maximum transmission limits over the channel [40]. This chapter presents the basic elements and provides a brief introduction to the general requirements of a digital data transmissiOn system operating over the telephone network. Particular emphasis is given to the derivation of the baseband model of QAM systems, employmg linear coherent demodulation at the receiver. This model reduces the cost of buildmg and testing the hardware of the new or improved digital commumcation system by allowing the system to be simulated using a digital computer. 2.2 GENERAL MODEL OF THE DATA TRANSMISSION SYSTE:M The system model under investigation

IS

as shown in Fig. 2.1. The data

communication system may be a serial or parallel system [5]. A serial system is one in which the transmitted signal compnses a sequential stream of data elements whose frequency spectrum occupies the whole of the available bandwidth of the transmission path. A parallel system IS one m which two or more sequential streams of data signals are transmitted simultaneously, and the spectrum of an md1vidual data stream normally occupies only a part of the available bandwidth [3,5,16,27] In a serial system the signal elements are normally transmitted at steady rate of a giVen number of element per second (baud). The receiver extracts the element tumng mformation from the received signal and operates m time synchronism with the received signal. Such a system is called a synchronous serial system [5]. A serial data transmission system is less complex than a parallel data transmissiOn system

7

as the latter needs several demodulators to process the different signals [5,12]. In application where a relatively high transmission rate is reqmred over a given channel, a synchronous serial system is the most commonly used system [5,12]. Therefore, it will be assumed throughout this work that the data transmission system is a synchronous senal data transmission system. A sequence of regularly spaced weighted impulses I.s,o(t -1T) representing the

' transmitted data signal elements are fed to the input of the transmitter filter with a baud rate of 1(f. O(t) is the dirac function and S, IS the

lth

transmitted data symbol

which may have m possible values;

s, = 21-m+1

1=0,1,

.

,m-1

21

The {s,} are considered to be statistically independent symbols and are assumed to be equally hkely to take on any one of m possible values in any one symbol period. The transmitter filter limits the spectrum of the transmitted signal energy to the approximate available bandWidth of the transmission path. The receiver filter removes the nOise components outside the frequency bandwidth which approximately corresponds to the bandwidth of the received signal [5,12,35]. The transmission path could be either a lowpass channel, with a frequency lirrut no greater than 10 kHz or a typical voice-frequency channel with a frequency band no wider than about 3 kHz such as could be obtained over the telephone network or an HF radio hnk [3,5,12,16,27]. In the latter-case, the transmission path in Fig. 2.1 is assumed to include a lmear modulator at the transrrutter and a hnear demodulator at the receiver. The transmitter filter, the transmission path and the receiver filter m cascade are assumed here to form a lmear baseband channel whose Impulse response IS y(t); for practical purposes, y(t) is assumed to be of finite duration and time mvariant over the mterval of any transmission. The only nOise mtroduced by the channel Is addltlve noise which can, for practical purposes, be taken to be additive white Gaussian nOise [3,5]. It has been shown that If one data transmission system has a better tolerance to additive white Gaussian

8

-----------------------------------

-

noise than another, it will also, in general, have a better tolerance to other type of additive noise introduced over telephone circuits [3,5, 12-13]. Furthermore, the effects of addmve white Gauss1an nmse on a digital data transmisswn system may readily be analysed theoretlcally and stud1ed by computer simulation. For these reasons, it will assumed that the only noise introduced at the output of the transmission path in the model of the data transmission system is additive white Gaussian noise. The nmse has a zero mean value and a two-sided power spectral density of ~N0 , giving the zero mean Gauss1an waveform w(t) at the output of the receiver [5]. Thus the output signal from the baseband channel in Fig 2.1 is the waveform [17]

=

r(t)

L:s,y(t- iT)

+ w(t)

2.2

where {s.} are the values of the transmllted data elements, y(t) is the impulse response of the equivalent base band channel, and T 1s the symbol duratwn. The continuous signal, r(t), at the output of the baseband channel is sampled, at the baud rate, to give the received samples {r,}, where 1 takes all possible integer values. Various techniques for holding the samplmg instant correctly synchronised to the received signal are given elsewhere [41-42]. The i'• rece1ved sample is therefore; g

r,

where r,

= h=O 2: s,_.y. =r(zT),

Y•

+ w,

2.3

=y (hT), w, =w(zT) and Y•

is the (h

+ 1)1• component of the

sampled 1mpulse response of the baseband channel. The delay in transrmsswn, other than that mvolved m the time d1spersion of the transmllted signal, is neglected here, so that y 0 "' 0 and Y• = 0 for hg. The sampled impulse response of the lmear baseband channel, or at least an accurate estimate of lt, 1s assumed to be known at the receiver and 1s given by the (g+ I)-component row vector, Y, which is g1ven by,

Y

=

[

Yo Yt

· Y,

9

]

2.4

In the signal processor and detector m Fig. 2 1, the values of the {.f,} are determined from the {r.} using any one of several different detection and equalization techniques. The signal processor IS assumed here to have prior knowledge of both Y and the possible values of the s,. The detected data value of s, IS designated to be §,

2.3 ASSESSMENT OF THE DISTORTION PRESENT IN THE SA:\IPLED IMPULSE RESPONSE OF A BASEBAND CHANNEL Two types of distortion can be present in a sampled waveform; these are amplitude and phase distortion [3,5,10-13,27]. For no amplitude distortion, all the amplitudes of samples at each frequency are scaled by a constant amount [5,10-12]. Therefore, the attenuation is independent of frequency. For no phase distorllon, the wave velocity must be linear [1 0-12]. Therefore, a graph of phase against frequency must be a straight line. Group delay is a common measure related to the phase distortion [5,12]. It is defined as the rate of change phase with frequency [5,10-12] Phase d1stort10n can be defined in terms of the components of the DFT (D1screte Fourier Transform) of the sampled impulse response of the correspondmg channel. When all the components of the DFT of the channel vector Y have all the magnitudes equal to unity, the channel represents pure phase distortion [10-12]. The deviation of these magnitudes from unity can be used to esnmate the level of amplitude distortion present Alternanvely, the aperiod1c autocorrelation functiOn has all components equal to zero except for the (g + 1)'• component, which has the value of unity [10-12]. These properties sho_w that pure phase distortion represents an orthogonal transformation [12,27]. Therefore, a suitable detecnon process provided at the receiver can reverse the orthogonal transformanon, phase distortion is not usually considered when assessing the distortion present in the sampled impulse response of the channel [27]. Amplitude distortiOn is a much more serious consideration because it always lowers the tolerance to noise of a data transmission system [12,27]. In terms of the components of the DFT of Y, the magmtude of the components are not equal to umty. Another important feature of amplitude distortion

IS

that it always changes

the discrete energy-density spectrum and the apenod1c autocorrelanon function of

10

the sampled impulse response of the channel [12,27]. The d1screte energy spectrum IS

the DFr of the a periodic autocorrelation function [10-12]. Pure amplitude

distortion means no phase distortion or delay. In order for the latter condmon to be sat1sfied, the components of DFr must be all real-valued [12,27]. These properties show that pure amplitude distortion is a symmetnc transformation [10-12,27]. In terms of the zeros and poles of z-transform of the sampled impulse response of the channel, ampl!tude and phase distortion can also be defined. When the channel introduces pure phase distortion, each zero and pole of the z-transforrn of Y are accompanied by a pole or zero, respectively, at the complex conJugate of the rec1procals value of z [27]. Whereas, when the channel introduces pure amplitude distortion means that the z-transform of Y contains an even number of zeros, each zero being accompanied by another zero at the complex conjugate of the reciprocal value of z [27].

2.4 TELEPHONE CIRCUITS Telephone circuits are an arrangement w!lh electrical interconnecting whereby the communicatiOn of speech or data can be carried between any two points. It may be pnvately or publicly owned. Pnvate circmts can be considered as point to point communical!on wh1ch are permanently or on a part time bas1s rented by one or more subscribers [3-6,43-46]. They are not connected through any of the switches in the exchange nor to the exchange or repeater stauon's battery supplies. Large organisal!ons such as banks and rarlway authonties may have their own network of l!nes to meet therr own demands [43] The publ!c sw!lched telephone network (PSTN) 1s necessary to connect any given subscriber to another at a telephone exchange (which may be manually or automatically operated) [35]. These services are most useful where the transmission time per day Will be relatively short, or when a central point has to communicate w!lh a large number of outstations [46]. The chmce between using private circuits or using the PSTN for data transffilssion must be made by careful consideration of factors, such as the costs, availability, speed of workmg and transmission performance. A PSTN connection

IS

establ!shed by

dmllmg the telephone number of the distant data terminal and the route used for a particular call1s a random chmce from a large number of d1fferent poss1ble routing,

11

includmg various combinations of audio-frequency cables and mulu-channel systems [45-46]. For data transmission, private-leased crrcuits have the following advantages over the PSTN [3-6,35,43-46]. The bandwidth ofpnvate circmts tends to be around 300-3000 Hz, v.h1ch can

i-

be adjusted to give a good performance over this bandwidth, whereas the bandwidth of the PSTN

IS

more restncted, nominally 300-500 Hz and

900-1200 Hz, to avmd the frequenc1es used for in-band signalling on truck routes n-

private circmts are less nmsy than PSTN channels because of the switching equipment used in telephone exchanges.

m-

Eff1cient performance due to exclus1ve use of the Circuit is obtained.

iv-

The link can be adjusted to have the optimum performance, where high speed and more reliable transmission are possible.

v-

Full-duplex operauon IS available at higher bit rates

Fmally, the cost of the permanently private circuit may be relatively high especially if there are insufficient data traffic on the line. In practice, most private data networks consist of a combinations of both leased and PSTN lmk. Vmce-frequency data circmts can also be of any length. Unloaded audio lmks are generally very short (of the order of 3 miles) [5]. They comprise a pair of balanced twisted wires with nominal impedance of 600 ohm. They have a good frequency response, w!!h some attenuation distortion and negligible delay distortion over the voice-frequency band [5]. Furthermore, and over a vo1ce-frequency as the length of the link m creases the attenuation increases which prevents the use of long unloaded audio lmks, where the mcrease in attenuation is about 2 dB per miles [5]. In many situations, It is deSirable to extend the length of the links beyond this limit of 3 miles. Common methods to attam longer links without exceedmg loss limits are the followmg [5,43-46];

12

i- Increase the conductor diameter. ii- Use amplifiers. hi- Use inductive loading. Loading a particular voice pair links consists of mserting series inductance (often 44 or 88 mH) into the lmks at a fixed mtervals (typically 2000 yds). Adding load coils tends to; i- decrease the velocity of propagation. Ii- increase the impedance. Over the centre of the vmce-frequency band the attenuatiOn distortion decreases due to the presence of the loading coils; in return the attenuation and group delay distortion will mcrease over the higher frequencies [5,43-46]. This mcreases as the length of the loaded audio link increases, especially at the htgh frequency end of the voiceband [5]. The third type of link is the carrier link which can be very much longer than loaded audio links. The modulation process at the transmitter is a single stde band suppressed-carrier amplitude modulation which shift the stgnal frequency band to some higher frequencies [5]. Because of the high cost of !me plant it is desirable to use a line carry more than one data lmk (multi-channel) by usmg a multiplexer. With frequency divisiOn multiplexing, each data channel is shifted or frequency translated to a different part of the available frequency spectrum. The particular frequency to whtch the channel is shtfted IS determmed by the frequency of the carrier whtch is modulated by the data signal. This combination of modulation and frequency division multiplexmg is the basts of the multi-channel carrier telephony systems which operate over carrier pairs, coaxial cable, microwave radiO relay systems, satelhte systems and HF radio links The distortions introduced m earner ctrcmts are entirely determined by the filters involved in the modulatiOn, demodulation and multiplexmg processes which originate at the termmal station and do not depend on the length of the lmk [5,10,35]. The correct operation of a multt-channel system rehes upon a earner bemg re-inserted at the receiver at the correct frequency.

13

Although elaborate synchronization circmts are used, the frequency offset may be as much as± 2Hz [5,46]. This leads to signal impairments rather than pure signal distortion.

2.5 ATTENUATION AND GROUP DELAY DISTORTION Signal distortion is the change in shape of the transmitted Signal, resulting from attenuation and group delay characteristics. These two charactenstics are the most widely studied characteristics of telephone connections and can be defined as the variation of the attenuation and group delay with frequency [5,10-12,35]. The attenuation distortiOn in a given frequency band is the variation of attenuation over that frequency band and group delay d1storuon !S the vanauon of group delay over the frequency band A switched line is made up of a number of separate lmks, each selected on a purely random basis from a large number of poss1ble routings [5]. It 1s therefore poss1ble to have echoes or reflected signals, if there 1s any mismatch m the system. This mismatch occurs because it is not possible to check and correct the frequency characterisucs of the complete circuit [5,27]. Such echoes can be extremely annoying and will worsen with mcreased delay. When there are two or more m1smatches along the !me, the received signal element comprises the mam components of the element followed by several echoes, which are attenuated and delayed with respect to the main component. Th1s effect of m1smatch is an example of multipath propagauon, where the received signal arrives at the receivmg end over more than one path. Echoes in the received signal mdicate the presence of the appropnate combination of attenuauon and group delay distortion, where the signal distortion 1s dependent upon the relauve magnitudes of echoes and main Signal components. Echoes can occur on bo.th sw1tched and pnvate circuits although because of the possibility of !me mismatch 1s greater for connecuons over the switched network, severe echoes are more likely to be encounted over the PSTN. F1g. 2.2 shows the attenuation and group delay charactensucs of an ideal voice-frequency channel, where the attenuauon and group delay nse rapidly at frequencies below 300 Hz and above 3400 Hz [5]. Channels introducmg severe attenuation and group delay distortion may have sharper responses [5].

14

-------------------------------------------------------------------------------------1

The attenuation and group delay characteristics causes time dispersion of the received signal which usually mcreases with the attenuation and group delay distortion in the signal frequency band [5,10-12,27,35] The effect of this time dispersion IS to spread out, in time, the response of a data pulse such that It overlaps the adjacent pulses in a digital wave train This IS called the intersymbol interference and reduces the margm against noise or, m severe cases, causes systematic errors [27]. The major effect of time dispersion is to set up an upper limit on the rate at which the element signal transmitted for an acceptable level of mtersymbol interference [35]. Therefore, channels with low levels of attenuation and group delay distortion will be able to transmit the data element at higher rates than those exhibltmg higher attenuation and group delay distortions. As a result, the impulse response of the channel (the signal at the channel output when the mput signal is the dirac function) IS a contmuous rounded waveform of duration not less than fraction of milli-second [10]. When the input signal to the channel is a sequence of impulses {o(t -zT)}, where i=1,2, ... , and T is the time interval separatmg two successive impulses, the output signal IS a sequence of continuous rounded waveforms, each being given by an appropriately delayed version of the channel impulse response. When the timeT is shorter than the duration of the channel impulse response, then consecutive output rounded pulses overlap producing what IS known as intersymbol interference [5]. In practice, a good channel will have an impulse response which a rapid rise to Its peak value followed by a rapid decay [5,10-12,27 ,35]. 2.6NOISE The term noise is used to designate i.mwanted signals that tend to disturb the transmissiOn and processmg of signals in commumcation systems. Telephone circmts mtroduces different types of additive and multiplicative noise [5,47]. Impulsive noise appears in the public switched telephone network as predommant additive noise because of the electncal/mechamcal switches in the exchanges [3-5,10-12,43-46]. Unfortunately, the shape of the impulsive noise varies widely from one telephone circuit to another, with durations extended over many adJacent signal elements, makmg the simulation of this noise by computer a difficult task [5,43-46]. Recently, electncal/mechamcalswitches have been replaced by electronic switches, which greatly reduce the impulsive nmse. Speech and signallmg tone

15

cross-talk is another type of addilive noise wh1ch occur because of capacitance unbalance between the pairs of wires used in a connection [5,43-46]. They are not normally important in causmg errors except at low signal levels or under !me fault conditiOns [5,46]. Wh1te noise, whose spectral density function is flat out to frequenc1es well beyond those occupied by any message bearing s1gnals under consideration, produces errors only at very low Signal level and is not normally a s1gmficant source of error [5,47]. The best examples are thermal nmse and shot noise which have a Gaussian amplitude distributions and are known as Gauss1an wh1te noise [47]. Passing this noise through bandlimiled telephone circuits w1ll produce band-limited while noise or coloured noise [47]. Strictly speakmg, white noise has an inf1mte average power and, as such, it

IS

not phys1cally realizable [47-49].

Nevertheless white nmse has convenient mathematical properties and therefore is useful in system analysis [3,5,10,35,47-49]. Any two different samples of while nmse are uncorrelated [5,12-13,16,47-49]. These samples are statistically independent if the white noise

IS

Gaussian d1stnbuted, because Gaussian nmse

represents the ultimate m randonmess [47-49]. The multiplicative noise involves both amplitude and frequency modulation effects [5,16]. The noise waveform amplitude or frequency modulate the signal waveform. The noise originate on earner lmks

and can be of several types as follows

[3,5,16,43-46]; i- Modulation noise. ii- Transient interruptions. iii- Sudden level changes. iv- Frequency offsets. v- Sudden phase changes. v1- Phase Jillers. Fmally, the majority of noise introduced over the switched telephone network 1s additive. Multiplication noise can be the predominant type of noise introduced over private lines [5].

16

-----------------------------------------------------------------------------

2.7 MODEL OF THE DATA TRANSMISSION SYSTEM USING QAMSIGNAL The model of the digital data transmission system usmg QAM (Quadrature Amplitude Modulation) signals is Illustrated in Fig. 2.3. The QAM system involves the transmission of two parallel signals each requmng a separate amplitude modulator at the transmitter and a separate amplitude demodulator at the receiver [11,27-29]. The two modulated signals have the same carrier frequency but are in phase quadrature. A QAM system is used here because it has several important advantages over other modulatiOn techniques, these are as follows, [10-12,26,35] 1-

Maximum avmlable tolerance to nOise achieved when used wah near optimum detection process.

ii-

Full use of the available frequency band, since the carrier frequency IS at the centre of the available frequency band.

iii-

S1mple equipment filters, since the amplitude and phase characteristics of the filter do not have to have any particular shape to satisfy any very stringent conditions, which is the case in the S S.B (Single sideband modulation), V.S B. (Vestigial sideband modulation) and I.S.S.B. (Independent smgle side band modulation) systems.

1v-

No p1lot earner needs to be transmitted with a QAM signal, since the rate of change of the relative phase angle is farrly small. This reduces the complexl!y of receiver by avoiding the ISohition of the p1lot earner from the data s1gnal at the receiver

The information to be transmitted IS a sequence of binary digits {Xk}, where Xk takes any of Its two poSSible values, 0 or 1. These binary d1g1ts are coded into two multi-level signal s•.• o(t -1T) and sb,,6(t- iT) The real and 1magmary parts of the correspondmg s, are statistically mdependent and equally likely to have any of their m poSSible values, where,

17

s•.• ,

sb,,

= 2/

-

m

+ 1

l

=

0,1,2,

.. ,

m-1. ..

2.5

so that any two data symbol (s.,.,sb,.) may have one of m 2 possible combmations. It has been assumed that, when the bmary digits are fed to the encoder at a rate of 9600 bit/s, the possible values of s•.• and sb,• are± 1 and± 3. It is also assumed that, when the bit rate is 19200 bit/s, the possible values of s••• and sb,, are± 1, ± 3, ± 5 and±7. Each of the two data streams s•.• and sb,• are fed separately to one of the lowpass filters at the transmitter, to be shaped to the appropnate bandwidth. These two lowpass filters have identical impulse responses, a(t), with transfer functions A(f). The output signals from these two filters are modulated by two carriers, m phase quadrature, but with the same earner frequency fo. The factor ..J2 in Vlcos(2nfor) and Vlsin(2nfor) gives each of these signal a mean-square value of unity [ 10,17 ,35,48]. The output of these two lmear modulators are added to form the QAM signal which is given by;

x(t)

= fi:I.s.,a(t -zT)cos2~J;t + fi:I.s •. ,aCt -zT)sin2~/,t '

.

'

26

The QAM signal is fed to the bandpass filter at the transmitter. This bandpass filter, for practical systems, is very necessary and is used to remove spurious frequency components generated in the modulator [10,17,35,48] It has an impulse response of g(t) with aFounertransfonn ofG(f). The resultmg QAM signal is then transnutted over the transmission path which has been assumed here to be a telephone circmt. The telephone crrcmt mtroduces linear amplitude and phase distortiOn into the transmitted signal, it has been descnbed in Section 2 3 with an impulse response of h(t) and transfer function of H(f) The only additive n01se assumed here is stationary white Gaussian noise, n(t), with zero mean and two Sided power spectral density ~N0 , which added to the QAM signal at the output of the telephone circmt [10-12,17,27,35,48].

18

At the receiving end, the linear demodulator includes at its mput a bandpass filter, the absolute value of whose transfer function matches the amplitude spectrum of the QAM signal at the input to the telephone circmt. The bandpass filter prevents over-Ioadmg of the two following mult1phers by noise in the received signal [48]. Th1s IS performed by removing the noise components outs1de the frequency band of the signal without excessively distortmg the signal Itself. It has an 1mpulse response of c(t) with transfer function of C(f). The output signal from the bandpass f!lteris now coherently demodulated by two reference earners wh1ch have the same frequency but are in phase quadrature. The two lowpass filters after the demodulator suppress the high frequency components so that only the baseband s1gnals are retained. These two Iowpass filters are as those in the transmitter with an impulse response ofb(t) and transferfunctwn ofB(f) [10,17,35,48]. It Will be assumed that the transmitter and rece1ver Iowpass filters are such that;

=

A(f)

=

B(j)

0

lfl > ..t;

2.7

and that ..t; is such that;

1

~

2.8

2T

where 1fT is the Signal-element rate. Defining the complex data symbols, as,

...

where J =

x(r)

H, Eqn =

29

2.6 may be wntten as

"'(se -J2•f,t _1r;;;"--v2 J

+ s , eJ2V,')a (t -11"") I

J

19

...

2.10

-----------------------------------------------------------------------------

where s.' is the complex conJugate of s,. The input signal to the coherent demodulator in Fig 2 3 is now given by

=

m(t)

x(t)



g(t)



h(t)



c(t)

+ n(t) * c(t)

...

211

where *indicates the convolutiOn operation. In the demodulator, the muluplicauon process causes the transfer function of m(t) to be shifted in a negative direction by fc Hz. The lowpass filters b(t) then remove the high frequency components leavmg

only the baseband components. The signal at the output of the two lowpass filters, b(t), are given by;

r 1(t)

=

[Vlm(t)cos(2nfct

+ 6)]

*

b(t)

r 2 (t)

=

[Vlm(t)sin(2nfct

+ 6)]

*

b(t)

...

2.12

213

where 6 is relative phase error between the modulator and demodulator caniers. Combining these two signals in a complex form will give the complex signal r(t)

=

r(t)

r 1(t)

+ j r 2(t)

. .

2.14

consequently, from Eqns. 2.10-2.12

LS1{a(t - iT)*[h(t)*g (t)*c(t)ei2Xt,r}*b (t)

r(t)

I

+ Ls,'{e'••t,ra (t - zT)*[h(t)*g(t)*c(t)ei2XI.r]e'•*b(t) I

+ -V2[(n(t)*c(t))e'""'·'+Ol]*b (I) .. 2.15

LS1y(t- zT) + w(t) I

where

y(t)

{a(t)

*

[(h(t)

* g(t) • c(t))e'~''"]e'' * b(t)

20

2.16

and

w (t)

==

-·~L[_ r;;f(n (t )

*

c (t))e 1 becomes [27]

000

where Er is the transpose of E and u, +•

IS

3.39

a Gaussian random variable with zero

mean and variance

112

= 2criDI 2

000

3.40

Let

r - g----.

n

E.

=

[0

0

0

0

1 0

0

0

0

0]

... 3 41

Now, since the data symbols {s.} are statistically mdependent with zero mean and vmance of 2 c? and since the noise components u, +• and any of the data symbol are statistically independent, it follows that

E[s,s)

=

for

0

i >"j

000

3.42

and

- for

any

j

3.43

Thus, from Eqns. 3.35-3.43 [27]

e

= 2o2IBI 2 + 2criDI 2

where B IS the (n+g+ I)-component row vector given by,

52

...

3.44

-

=

B

E

-

F0

-

E.

...

3 45

It is clear from Eqn. 3.44 that the terms 2821B 1 2 and 2d'l D 1 2 are the mean-square error due to intersymbol mterference and noise, respectively, and the equalizer must be adjusted to minimize both errors. It can be seen from Eqns. 3.44 and 3.45 for any giVen

value

of

J; =e•• , for

D,

2 821B 12

and

e

are

minimized

by

setting

J=I,2, ... g.

Let I be an (n+ I)-square identity matnx and

z. be an (n+ I) upper triangular square

matnx

,, ,,

,,

Yo

0 0

0

'· '·-·

y,

,,

,,_,'·

J,-J

z. ;

0

0 0

0 0 0



345

0 0 0

0

,,

0 0

The (i + I)'• row of

0 0

z.

0

,, ,, ,, y,

0

is derived from the first row by shifting its non-zero

components 1 places to the right, discarding all components after the (n + I)'• and setting the first i components to zero. Fmally, let transpose of

z•.

z; be the complex conjugate

It has been shown [27], that the (n+I)-tap linear feedforward

transversal filter, D, of the deciSlon feedback equalizer of Fig. 3 5 that mimmizes the mean-square error in the equalized signal (Eqn. 3 44), has tap gams given by the (n+ I)-component row vector [27 ,62]

D

=

( E.z:lz.z:

d' + 02 /

)-!

53

...

3.47

~

-------------------------------------------------------------------------------

and the tap gains of the filter F being given by Eqn. 3.34. The matrix

z.z: + ( ~

J

is an (n+ l)x(n+ 1) positve-definite Hermlt!on matrix, bemg posllive-defmite so long as ri' ;e 0 or

z.z; IS non smgular and therefore it has an mverse [63,64]. From Eqns.

3.44 and 3.47, the mean-square error in the equalized signal becomes [27],

...

3.48

3.4.3 Comparison of Equalizers a: In the presence of phase distortion When a channel mtroduces pure phase distortion and IS physically realizable, each pole and zero must be accompanied by a zero or a pole, respectively, located at the complex conjugate of the reciprocals of their values [12]. Therefore, the z-transform of the sampled impulse response of the channel can be represented by

Y(z)

~

Y2(z)y;- 1(z)

...

3.49

where Y2 (z) and Y3 (z) are given by Eqns. 3.27 and 3.31, respectively. Therefore, the z-transform of the filter D of the ZF equalizer is now given by

D (z)

~

z -•y;-'(z )Y3(z)

...

3.50

...

3.51

The z-transform of the channel and filter therefore becomes

Y(z)D(z)

~

z-•

Thus the sampled impulse response of the channel and hnear filter D IS

,--g---.,

n

E

~

[0 0 . . . 0 1 0 . . . 0]

54

3.52

--~

where n and g are ideally infinite but, for practical purposes, can be taken to be appropriately large positive integers. It is clear from Eqn. 3.52 that the tap gains of the filter F are now all zero which means that the decision feedback equalizer has degenerated into a linear equalizerthat achieves the exact equalization of the channel. Furthermore, since the sequence w1th z-transform f 3(z) IS the complex conjugate of the reverse of the sequence wtth z-transform Yiz ), 1t can be shown that the sequence "of the reverse

with z-transform z-"f21(z)Y3(z) is the complex conjugatelof the sequence wtth z-transform Yiz )Y;1(z ), appropriately delayed [27]. Clearly, the filter D is not only the inverse of the channel but IS also matched to the channel, its tap gains being the complex conjugates of the channel sampled impulse response, and m the reverse order. Thus the ZF equalizer is the optimum detector for the given rece1ved signal. When the channel mtroduces pure phase d1stortion, the squared matnx

z. in the

MMSE equahzer, Eqn. 3.46, must ideally have an infmite order [62]. However, a very good approximation to the ideal can ach1eved in practice by a matrix w1th a finite order. It

IS

clear from Eqn. 3.46 that the last column of z.

IS

the sampled

impulse response of the channel, as given by the fust row of z•. but in the reverse order. The square of the unitary length of th1s column is

IYI 2 =

1

3 53

It has been shown that a sequence representmg pure phase distortion IS orthogonal to Itself shifted by any non-zero integral number of places [27], so that the inner product of two sequences is zero. It follows from Eqn. 3.46 that the last column of

z. is for practical purposes orthogonal to each of the other n columns. The factor

E.z; in Eqn. 3.47 is an (n+ 1)-componentrowvector given by the conjugate transpose of the last column of Z., so that E.z;z. IS the (n+ I)-component row vector whose i'h component (fori= 1, 2, .... , n+ 1) is the mner product of the ith and (n + 1)'h columns of z•. This must be approximately zero, except when i=n+ 1, when it is unity (from Eqn 3.53). Thus

E.z:z.

n ~

[0

0

0

0

1] '

= E.

3.54

55

-j

----------------------------------------------------------------------------

and

...

It follows that

E,z;

3.55

is an eigenvector of the matrix z.z;, assoc1ated with an

eigenvalue ofumty [62]. Hence

E.z.·(z.z.• + ci I? I )

~

...

3.56

so thatE.z; is an eigenvector of z.z; + (~J associated with an eigenvalue of [62] 1 +(~).From Eqn. 3 56,

E.z.•

~

(1 + ci1, ·( . ci 82 f.z. z.z. + 82 I

)-I

...

3.57

...

3.58

or

(1

~

ci)-lE.z..

+ 82

which means that E,z; 1s an eigenvectorofthe matrix [z.z; +(~}J associated 1

1

wllh an e1genvalue of[1

+(~]- • From Eqns. 3.47 and 3.58, the tap gains of the

filter D are now given by

56

-

D

(I

r;~)-1

.

+ 02 E,Z,

3.59

As the signal-to-noise rauo becomes extremely htgh, r;l- ---7 0 and D ---7 E,z;, which means that the filter D becomes the same as for the ZF equalizer. When cr'- 1' 0, the tap gains of the MMSE equalizer are [I + ( ~ ]- times those of ZF 1

equalizer, so that the magnitude of the equalized stgnal is reduced by the same factor. This introduces a bias m to the equalized signal such that the real and Imaginary parts of the data component of x, are no longer symmetrically placed with respect to the decision thresholds. Inevitably, this degrades somewhat the performance of the equalizer. b: In the presence of amplitude distortion

In the presence of amplitude distortion it is more difficult to compare theoreucally the performances of the MMSE and ZF equalizers. Thts is essentially because the ZF equalizer is no longer an optimum detector, so that the MMSE equalizer may or may not now have a better tolerance to noise. The fact that the mean-square error in the equalized signal of the MMSE equalizer must be smaller than that m the ZF equalizer, together with the fact that the ZF equalizer has no mtersymbol Interference in the equalized stgnal (with the correct detection of the {s.} ), necessanly implies that the mean-square value of the noise _in the MMSE equalizer is smaller than that in the ZF equalizer. Indeed, since the mean-square error in the MMSE equalizer is caused partly by noise and partly by mtersymbol interference, with a natural tendency for the two be present at rather similar levels, it seems that the equalized Signal of the MMSE equalizer should normally have a sigmficantly lower noise level than that of the ZF equalizer. However, the advantage gained here by the MMSE equalizer is offset by the presence of intersymbol interference m liS equalized signal. The effect of the latter on the tolerance to the nOise is not easy to evaluate theoretically, butts best determined by computer simulation tests. The results presented m chapter 4 and 6 clearly demonstrate this effect.

57

A

(s, l

(x,)

.t.(r,)

Lmear equahzer

Detector

Fig. 3.1 Receiver using linear equalizer

r,

T

T

r, 2

.,.._;:=--··--·······

T

r •-n

.----L------I----I------····--·····--.t._...., L-

x,

'--------------···············----'

Fig. 3.2 Linear feedforward transversal equalizer for the baseband channel

58

A

s,

Detector I

Lmear feedforward transversal f:tlter

Yo

Fig. 3.3 Receiver using nonlinear equalization by decision-directed cancellation of intersymbol interference

r, Yo

A

s,

x,

Detector

A

A

T

T

T

Yo

Fig. 3.4 Detector and pure nonlinear equalizer

59

A

r1

T

r1.1

T

r1-2

T

r 1-n

X

St-n

1

Detector

q1

dn

X

A

A

A

S1-n-g

St-n-2

St-n-1

T

T

"'

T

0

Linear filter D

Lmrear filter F

----------------

Fig. 3.5 Decision feedback equalizer

-----------------

Channel! Real 10000 0 5031 -0 1447 00300 00094 -0 0136 0 0102 -0.0108 00083 -00044 00001 00014 -0.0005 -0 0007 00000 00004 -00002 -0 0002 -00002 00000 00000 00000

Channel2

Imagmary 00000 02008 -0.0083 -0 0097 00077 -0 0039

1.0000 0.5006 -0.1678 0.0176 -0 0062 -0 0146

00006 -0 0013 00024

00080 -0 0049 00063 -0 0031 00013 00012 -0 0003 -00011 0.0004 00008 -00002 00004 -0 0002 00004 00001 00000

-0 0020 00024 -0 0021 00019 -0 0008 -0.0002 00006 -0 0004 -0.0003 00004 0.0003 0.0000 00000

10000 04608 -0.5824

00007 -0 0011 -00002 00003 00000 00003 00000

00013 -0 0016 -0.0003 -0 0001

----

----

-------

----

----

----

----

----

-------

----

----

----

Real

00000 0.3397 00282 -0.0391 00229 -0 0132 00012 -00004 -0.0001 00027 -00020 -00003 00001 00002 -00008

----

----

Table 3.1

Imagmary

----

----

-------

Real

Channel3

0.1573 -0 0175 -0 0021 -0 0021 -0 0051 00080 -0 0039 -0 0001 00038 -0 0009 00023 -0.0004

00003 -0 0007 -0.0008 00001 -0 0001 -0 0001 00000

----

Channel4

Imagmary 00000 1 1004 00436 -0 1729 00872 -0 0194 00083 -0 0075 00054 -0 0035 00014 -0.0056 00026 -0 0027 -0 0009 00018 -0 0013 00000 00001 00002 00011 -00003 00002 00007 00004 0.0001

----

Real

Imaginary

10000 02458 -1.7189 06743 00356 -0.1155 00296 -0 0168 00160 -0 0146 -0.0002 00020 -0.0003 -0 0025 00066 -0 0035 -0 0061

00000 1.9797 -0 2028 -0 7923 05062 -0 1449

00059 -0 0008 -0 0006 -0.0013 00010 00015 -0 0003 -0 0001

-0 0009 0 0038 -0 0002 -0 0003 -0 0024 0 0009 0 0005 -0 0001

00000 00000

00000 00000

00385 -0 0370 00185 -0 0011 -0 0029 -0 0040 0.0002 -0 0074 0 0044 00050 -0 0028

Sampled Impulse responses of the mimmum phase versions of channels

1-4.

61

Channel6

ChannelS Real

Imagmary

1.000000 1.346393 -0 148268 -0222296 0.200053 -0.139489 -0.029114 0 040268 -0 006336 -0 005842 0022487 -0 020116 0010094 -0 000049 -0 014825 0022883 -0 024204 0 017227 -0 005489 -0 007162 0.015331 -0 013887 0 007161 0 001364 -0 004231 0 005047 -0 003534 -0.004201 0.000481 0000392 0000055

0000000 0 359591 0 330302 -0 087503 0 002161 0036076 -0 019378 -0 011521 0 019953 -0 010060 -0005228 0018909 -0 021531 0022504 -0 019873 0.008106 0003678 -0 014030 0.018100 -0 016173 0 008199 0 000381 -0 008364 0 009576 -0.005552 -0 001576 0004075 -0002520 -0 001848 -0.000019 0000103

----

-------------

----------

----

-------------

-------

Table 3.2

----------------------

Real 1000000 1.173137 -0.169015 -0.138032 0 199112 -0 117248 -0004179 0 043908 -0 021240 0 012810 -0006241 0002090 -0.000932 -0002066 0001822 -0 001417 0.001212 -0000042 -0000445 -0 001481 0 001702 0003671 0.004560 -0011131 -0 008303 -0 003463 -0 000987 -0 000218 -0000024

-------------

-------

----------------------

Channel?

Imagmary 0000000 0202423 0.108631 -0 049378 0 027682 0.007974 -0 009518 0006603 0001454 0002679 -0 004201 0.002607 0 003301 -0 005158 0 002641 0.001506 -0 003261 0002869 0 000283 -0.004083 0 002728 0002814 -0 002912 -0 000303 -0.000210 0 000532 0000288 0 000112 0000016

----

----------------

----------------

-------

Real 1000000 1.220912 -0 849086 -0452288 0482880 -0 256621 0 000383 0 096231 -0.070512 0024782 -0 002350 -0 006643 0005802 -0 004082 0003905 0 002613 -0 001821 0 007187 -0 003219 0.003162 -0 000893 -0 001047 0000104 -0 002343 0002986 -0 002837 0001167 0000222 -0003119 0 002629 -0.001428 -0 002190 0002643 -0 000528 -0 002267 0 002710 0 001723 -0 001346 -0 002851 -0 001450 -0 000571 -0000072

Imagmary 0000000 1339438 I 089636 -0 648090 0 037754 0212632 -0 201586 0 091226 0.008677 -0 021441 0 016130 -0 003623 -0 001010 -0 000398 -0 005003 0002401 -0 003081 -0 002388 0 002566 -0 003157 0002685 -0 001343 0 000071 -0000349 0 000001 0 001728 -0 000987 0 001269 -0 000076 -0 000807 0 001386 0000206 -0 000385 0000282 -0 000265 -0 000468 0 000499 -0 000249 -0 001539 -0 001316 -0 000521 -0 000038

ChannelS Real 1000000 1416469 -3 199131 -2 316267 2442584 -0.246997 -0 681720 0621926 -0252233 -0 054037 0.091835 -0.040897 -0.003828 0 016616 -0 008264 -0.001113 -0 004185 -0 000335 0 016538 -0 000591 -0 007670 -0 001936 -0 000837 -0 000885 0.000102 -0.000102 0000088

·---

----------------

----------------------

-------

Imagmary 0 000000 2 774325 2.788691 -2 920049 -0 938336 I 629715 -0 732925 -0005074 0293966 -0 203288 0 044110 0032256 -0 027159 -0 003513 0 000749 -0 013205 0 001288 -0 006626 0 005191 0 012238 0 002827 0002042 -0 000467 -0.000958 -0 000414 0 000190 0 000130

----------

-------------------

----------

-------

----

Sampled impulse responses of the minimum phase versions of channels

5-8.

62

CHAPTER4

ADJUSTMENT OF THE PRE-DETECTION FILTER

4.1 INTRODUCTION This chapter is concerned with the adaptive adjustment of an advanced data receiver, when operating in the presence of signal distortiOn and additive noise. For channels introducing severe degree of bandlimiting, coupled with the requirement of a high transrmssion rate, a s1gmficantly better performance can be ach1eved by employing a linear f1lter, called pre-detection filter, ahead of the detector [27,29,32,35,61] as shown m Flg. 4.1. The function of the filter is to concentrate the energy of the sampled impulse response of the channel and filter such that it appears towards the earlier samples [56]. Furthermore, it rmmm1zes (according to some cnterion) the components (pre-cursors), proceedmg the largest component m the sampled impulse response of the channel and filter which cause mtersymbol interference to future data symbols [56,61]. Thepre-detectionfiltercan also be combined with anon-linear equalizer (Flg. 4.1), wh1ch reqmres the knowledge of an estimate of the sampled 1mpulse response of the channel and pre-detection filter, to remove the mtersymbol interference caused by previously detected data symbols This chapter investigates several methods to actually ach1eve the adaptive adjustment of the pre-detection filter by studying two adjustment cntena, that of max1mizing the s1gnal-to-noise ratio at the detector input when the tap-coefficients are g1ven by the last (n+ !)-coefficients of the infmite set of values used m the 1deal filter (where n is an appropriate integer) and secondly, the mmimum mean-square error criterion. Forthe adJustment of the f1lter according to the first cnterion, three different schemes have been presented. The linear feedforward transversal pre-detection filter is, here, an all-pass network with ideally an infinitely long sampled impulse response, that adjusts the combined sampled impulse response of the channel and f1lter to be minimum phase w!lhout, however, changing any amplitude distortion introduced

63

by the channel and without changing the Signal to noise ratio at the output of the filter, compared to that at its input [27]. The pre-detection filter can, forconvemence and as an aid to understandmg, be considered to operate in two stages. Firstly, It equalizes all phase distortiOn introduced by the channel to give a resultant sampled impulse response that IS lmear phase m character. Secondly, the filter converts the linear phase response into a minimum phase response which results m a sampled impulse response that has a rapid nse to its peak value, followed by a rapid decay, with relatively few post-cursor lobes; in essence, this ensures that the energy of the sampled impulse response of the system is concentrated towards the earher samples with especial emphasis on the first sample, without, however, changmg the signal to noise ratio at the output of the filter [35]. All three schemes require a knowledge of the roots of the z-transform of the sampled impulse response of the channel that he outside the umt Circle in the z-plane. These roots are then used to determine the tap-coefficients of the pre-detection filter such that the cascade of the sampled impulse response of the channel and filter has a z-transform whose roots lie ms1de the umt crrcle. The schemes differ by the mechanism used to determine the coefficients of the pre-detection filter, wh1ch for perfect adjustment, will be infimte in number Clearly, in practice, the number of taps must be restricted to the smallest number consistent with adequate equalizatiOn. The fourth and the final scheme presented differs from the first three schemes in that the adjustment cnterion attempts to minimize the mean-square error m the equalised signal. Computer simulation studies when operating over models of eight different telephone channels are presented for various schemes. The results are arranged to show the comparative accuracies and complexities (m terms of the number of arithmetic operations required to Implement each scheme). Furthermore, the results have been arranged to Illustrate the behaviour of the adJUStment schemes m the presence of addmve white Gaussmn nmse and different amount of distortion introduced by the channels, as well as the system dependence upon the length of the linear pre-detection filter.

64

4.2 ADJUSTMENT SCHEME 1 Let Y(z) be the z-transform of the sampled Impulse response of the lmear baseband channel, given by

Y(z)

...

where {u,} are the negative of the g roots (zeros) of Y(z), such that Y(z

4.1

=-u,) =0.

Y(z) can also be written as ...

4.2

...

4.3

...

4.4

where

with all zeros lying within the umt circle in the z-plane, and

with all Its k zeros lymg outside the unit crrcle. Thepre-detection filter, D, in Fig. 4.1, is a Iinearfeedforward transversal filter whose tap-coefficients are given by the (n+ 1)-component vector D

=

[d0

d1







dJ

...

4.5

...

4.6

with z-transform

65

The resultant sampled impulse response of the channel and filter is therefore the (n+g+ I)-component vector,

...

4.7

...

48

whose z-transform is, E(z)

=

Y(z)D(z)

+ . . . + en+g z-59, revealed no significant improvement in the values of 'Jf1 and 'Jf2 but mvolved a large increase in the number of arithmel!c operations mvolved. Finally, the values of the mean-square error in the equalized signal due to the presence of additive white Gaussian noise, together with the overall mean-square error introduced by the combined effect of addl!ive nmse and mtersymbol interference, have been computed. The mean-square error due to the addltlon of noise only is taken to be given by,

4.20

where D is as defined in Eqn 4.5 but where D is smtably scaled such that e. is equal to unity, and where 2r:f is the variance of the (complex) noise process, w,. Tables

69

4.7 and 4 8 show the results of these tests for various values of signal-to-noise ratio; the results were found to be independent of (n+ 1), the number of taps in the hnear filter D. The overall mean-square error present in the signal at the detector input, x, due to both additive noise and intersymbol interference is given by,

e =

IO!og10 (

1 20000 .~ lx, 20000 1

- s,_,i

2)

4.21

In tests fore, it is assumed that the signal passmg through the filter F (Fig. 4.1) is s, mstead of s,. The value of e for various conditions was obtained from computer Simulauon tests involving the traimng sequence of20,000 symbols; these values for different signal-to-noise ratios are tabulated in Tables 4.9-4.16, for values of n set to 29, 39, 49 and 59.

4.3 ADJUSTMENT SCHEME 2 As with scheme 1, this scheme attempts to determme the pre-detection filter tap-coefficients such that the sampled Impulse response of the channel and filter is minimum phase. Furthermore, the tap-coefficients, {d.}, of the filter depend only on the kroots ofY(z) thatlie outside the unitctrcle in the z-plane. Having determmed these roots (by using some suitable root-findmg algorithm), the adjustment scheme forms, as with scheme 1, the polynomials Y2(z) and Y3 (z), where Y2(z)

= Y2.o +

Y2.1z

Y3(z)

=

YJ,Iz

-1

+

+

Y2.kz

+

YJ,kz

-k

...

4.22

...

4.23

and

YJ,o

+

-I

+

Also it forms the polynomta!

70

. .

-k

==

.

Y. o + Y..,z

+ • · · + Y. _. 30

-18 54

-18 80

-15 47

-10 27

40

-28 54

-28 80

-25 47

-2027

50

-3854

-38 80

-35 47

-3027

60

-48 54

-48 80

-45 47

-4027

70

-58 54

-58 80

-5547

-5027

80

-68 54

-68 80

-65 47

-6{)27

Table4 7

Mean-square error m the equahzed s1gnal due to add1Uve wh1te Gausswn noise only usmg scheme I.

sr-..'R

ChannelS

Channel6

O!annel7

Cllannel8

20

254

070

8 22

18 41

30

-746

-9 30

-I 78

8 41

40

-17 46

-19 30

-11 78

-I 59 -1159

50

-27 46

-2930

-21 78

60

-37 46

-3930

-31 78

-2159

70

-4746

-49 30

-41 78

-31 59

80

-57 46

-5930

-51 78

-41 59

Table4 8

Mean-square error m the equahzcd s1gnal due to add1Uve whlte Gauss1an nmse only usmg scheme I.

85

----------------------------------------------------------------------------------------------

sl\ra

Channel!

Channel2

Channe13

Channel4

20

-8 58

-8 83

-5 51

-028

30

-18 52

-18 78

-15 45

-9 86

40

-28 52

-28 78

-25 45

-17 35

50

-38 55

-38 80

-35 44

-2007

60

-48 53

-4875

-4513

-2056

70

-58 47

-58 31

-5297

-2057

80

-'R

Channel!

Channel2

Channcl3

Channel4

20

-8 58

-8 83

-5 51

-() 30

30

-18 53

-18 78

-15 45

-10 21

40

-2853

-28 78

-25 46

-19 92

50

-38 55

-38 81

-35 48

-2771

60

-48 54

-4880

-45 47

-30 81

70

-58 53

-58 79

-55 46

-31 22

80

-68 55

-68 80

-65 48

-31 25

Table 4 13

Mean-square error m the equaliZed s1gnal w1th 50-taps m the hncar filter D, usmg scheme L

SI\'R

ChannelS

20

258

Channel6 072

Channel?

ChannelS

8 73

18 44

30

-7 29

-929

167

854

40

-15 91

-19 12

-() 61

-() 36

50

-2039

-2825

-() 88

-544

60

-2129

-33 66

-() 91

-6 51

70

-2140

-34 83

-() 95

-667

80

-2132

-34 95

-() 80

-652

Table 4 14

Mean-square error m the equal !Zed signal With 50-taps m the hnear filter D, usmg scheme I

87

SNR

Channel I

0Jannel2

Channel3

Channel4

20

-865

-8 89

-605

-3 03

30

-18 57

-18 83

-IS 57

-11 15

40

-28 54

-28 82

-25 49

-2037

50

-38 55

-38 80

-35 50

-30 19

60

-48 63

-48 88

-45 56

-39 89

70

-58 56

-58 84

-5549

-49

80

-68 58

-68 85

-65 51

-58 59

Table 4_15

os

Mean-square error m the equaliZed s1gnal with 60-taps m the lmear filter D, usmg scheme L

Sl\'R

Channel 5

Channel6

Channel7

ChannelS

20

2 57

072

8 39

18 44

30

-7 38

-930

-t 26

8 43

40

-16 77

-19 23

-8 62

-I 44

50

-23 26

-2925

-11 21

-1079

60

-2512

-3925

-11 58

-1711

70

-25 31

-4860

-1165

-18 79

80

-25 39

-54 81

-!I 52

-18 85

Table 4 16

Mean-square error m the equaliZed s1gnal w1th 60-taps m the lmear filter D, usmg scheme L

Cbarme12

Channel I

Channel3

Channel4

n+l

Add.& Sub

MulL

20

1077

708

1106

805

1106

805

1292

913

30

2342

1786

2371

1785

2371

1785

2557

1893

40

4107

3148

4136

3156

4136

3156

4322

3273

50

6372

4928

6401

4945

6401

4945

6587

5053

60

9137

7108

9166

7125

9166

7125

9352

7233

Table 4.17

MUiL

Add. & Sub

Add.& Sub

MulL

Number of anthmet1c operat1ons mvolved m scheme 2

88

Add& Sub

Mult

Channel6

ChannelS

ChannelS

Channel?

n+1

Add. &Sub

Mult

Add. & Sub

Mull

Add.& Sub

Mull

Add. & Sub

Mull

20

1292 2557

913 1893

1356 2621

950

1590 2855

1085

1682 2947

1138

4322

3273

4386

50

6587

60

9352

5053 7233

6651 9416

30 40

Table 4 18

1930 3310 5091

6885

7270

9650

4620

2065 3445

4712

2118 3498

5225

6977

5278

7405

9742

7458

Number of an!hmetic operatiOns mvolved m scheme 2.

Channell

Channel2

Channe13

Channel4

n+1

Add.& Sub

Mull

Add.& Sub

Mull

Add & Sub

MulL

Add.& Sub

Mull

20

735

492

980 1340 1780

656

980 1340

656 976

1960

1312

976

1780 2220

1296 1616

2680 3560 4440

1952 2592 3232

2900

1936

5800

3872

30

so

1005 1335 1665

732 972 1212

2220

1296 1616

60

2175

1452

2900

1936

40

Table4.19

Number of an!hmeuc operatiOns mvo!ved m scheme 3.

ChannelS

Channe16

Channel?

ChannelS

n+1

Add. &Sub

Mull

Add.& Sub

Mull

Add. & Sub

Mull

Add.& Sub

Mull

20 30 40

1808 2680

1312 1952

1804 2684

5335

3564

2925 4355 5785

2132

2592

1476 2196 2916

2695 4015

3560

2025 3015 4005

3172 4212

4440 5328

3232 3872

5095 6984

3636

6655 7975

4444 5324

7215 8645

5252 6292

50

60

Table4 20

4356

Number of anthmeuc operatiOns mvolved m scheme 3

89

Channel}

Channel2

Channel4

Charmel3

S~'R

w,

w,

w,

w,

w,

w,

w,

w, -{) 18

20

-3908

-3379

-39 95

-34 59

-27 48

-17 18

-1972

30

-58 56

-53 53

-59 41

-54 36

-43 89

-33 76

-3053

-9 31

40

-78 14

-73 52

-7678

-1514

-6278

-53 95

-43 56

-28 36

50

-88 57

-85 93

-8019

-65 28

-7248

-53 56

-50 14

-1255

60

-8903

-6387

-8030

-4515

-73 02

-33 20

-6069

-3 61

70

-8911

-43 92

-8084

-25 69

-74 52

-1470

-7079

-{) 91

80

-8976

-2457

-8093

-978

-8275

-2 94

-7736

252

Table4 21

Discrepancy between the mimmum mean-square error sequence and the mmimum phase sequence with 30-taps m the filter D.

ChannelS

SNR

Channel7

Channel6

w.

w,

w.

w,

Charmel8

w,

w.

w,

w,

20

-2005

-2 84

-2054

-5 98

-13 93

3 94

-11 02

1613

30

-29 05

-7 83

·29 4t

·15 22

-2136

-t 36

-t5 15

1449

40

-38 60

-15 08

-37 02

-14 67

-3090

-12 57

-2t 98

tl54

50

-4862

-2t 04

-47 16

-523

-4099

-8 83

-29 59

576

60

-5975

-t217

-57 32

-{) 87

-4995

-t 41

-3873

-tl 59

70

-6892

-736

-67 t7

I 19

-59 13

3 69

-4516

7 48

80

-7797

-4 93

-7766

309

-68 24

6 64

-54 81

14 99

Table4.22

Discrepancy between the mimmum mean-square error sequence and the mmimum phase sequence With 30-taps m the filter D. Olannell

Channel2

Channe13

Channel4

S/'."R

w,

w,

w,

w,

w,

w,

w,

w,

20

-39 07

-3379

-3995

-3459

-2749

-17 17

-19 72

-{) 18

30

-58 56

-53 53

-59 44

-54 34

-43 85

-33 58

-3055

-9 29

40

-78 51

-73 50

-79 38

-74 32

-63 10

-52 89

-44 89

-25 29

50

-9847

-93 51

-99 05

-94 47

-8292

-7286

-52 91

-19 59

60

-t16 07

-liS t5

-109 88

-t07 08

-9810

-9056

-61 57

-8 17

70

-11968

-105 50

-tlO 39

-85 42

-9978

-6702

-7229

-3 66

80

-tl9 74

-8525

-tlO 40

-65 41

-99 84

-47 98

-8665

-2 tl

Table4 23

Discrepancy between the mimmum mean-square error sequence and the mimmum phase sequence with 40-taps m the filter D.

90

Channel6

ChannelS

SNR

'V,

20

-20 06

-2 33

30

-29 36

.673

40

-39 61

50

ChannelS

Channel?

'V,

'1/,

'V,

-2070

-595

-13 93

3 94

-11 02

16 13

-3067

-13 82

-2136

-136

-15 19

14 49

-1241

-43 94

-26 85

-3090

-12.57

-22 21

11 68

-50 81

-2145

-52.85

-29 12

-4099

-8 83

-3073

6 84

'V,

'1/,

"'·

60

-5771

-18 91

-58 90

-13 40

-49 95

-I 41

-4020

-439

70

-69 rn

-965

.6515

-4 93

-59 13

3 69

-4825

0 85

80

-8159

.653

-77 80

.039

.68 24

664

-56 82

10 10

D1screpancy between !he mmimum mean-square error sequence and !he m1mmum

Table4.24

pbase sequence w11h 40-taps m !he filter D.

Channel2

Channell

S!\'R

'1/,

Channel4

Channel3

'V,

'V,

'V,

'1/,

'V,

'1/,

20

-39 rn

-33 79

-39 95

-3459

-27 50

-17 17

-1972

.0 18

30

-5856

-53 53

-59 44

-54 34

-43 84

-33 55

-3055

-929

40

-78 51

-73 50

-79 39

-74 31

.6306

-52 83

-45 76

-2419

50

-98 50

-93 50

-99 38

-94 31

-8217

-72 75

-59 81

-44 21

60

-118 5 0

-113 50

-119 35

-114 32

-102 94

-92 80

-64 53

-19 85

70

-138 2 t

-13373

-13691

-135 67

-12144

-115 60

-71 fl7

-1023

80

-1497 5

-145 29

-140 50

-125 65

-12665

-100 88

-81 07

-4 01

Table4 25

Dtscrepancy between !he mmtmum mean-square error sequence and !he mmimum pbase sequence WIIh SO-taps m !he filter D.

ChannelS

S!\'R

'V,

20

-20 06

Channel6

Channel?

ChannelS

'V,

'1/,

'V,

'1/,

'V,

'V,

-233

-2070

-5 95

-13 98

392

-1102

1613

30

-30 00

.672

-3067

-13 82

-21 71

-I 11

-1519

14 49

40

-4008

-12.14

-4425

-2649

-3171

-9 31

-22.22

1168

50

-5126

-20 65

.6027

-48 90

-4154

-17 42

-3097

696

60

-641 4

-4005

.64 83

-3167

-51 28

-4 73

-41 40

-150

70

-70 22

-21 38

.6715

-13 69

.62.39

.052

-5164

-15 99

80

-77 83

-1141

-76 31

-3 74

.6844

2 00

-59 24

3 12

Table4 26

Discrepancy between !he mm1mum mean-square error sequence and !he mmimum pbase sequence With SO-taps m !he filter D.

91

Channel2

ChaMell

Channel3

SNR

'If,

IV,

IV,

IV,

20

-39rrl

-33 79

-3995

-3459

30

-5856

-5353

-59 44

40

-78 51

-7350

-7939

50

-9850

-93 50

-99 39

60

-118 50

-113 50

70

-138 48

-133 58

80

-159 06

-153 65

Channe14

'If.

IV,

IV,

'If,

-27 50

-1717

-19 72

-0 18

-5434

-43 84

-33 54

-3055

-9 29

-7432

-6305

-52_81

-45 81

-2414

-94 31

-8296

-7273

-62 34

-5074

-11938

-11431

-10295

-9272

-66 88

-2727

-139 37

-134 34

-122 94

-112 72

-7286

-13 06

-159 67

-15147

-142 67

-133 96

-83 21

-7.58

Discrepancy between !he mmimum mean-square error sequence and !he mimmum phase sequence with 60-taps m !he filter D

Table4 27

ChaMelS

Channe16

Channel?

Cha!Ulel8

SNR

'If,

IV,

'If,

IV,

IV,

IV,

IV,

20

-2006

-233

-2070

-595

-13 98

392

-11 02

16.13

30

-29 37

-672

-3067

-13 82

-21 75

-I 10

-15 19

1449

40

-4009

-1211

-4429

-26.46

-3275

-8 53

-2223

1168

50

-5158

-19 78

-6219

-4438

-45 05

-2055

-3098

697

60

-6447

-3708

-80 68

-6697

-5514

-2074

-4177

-I 21

70

-73 05

-2733

-86 63

-54 91

-62 44

-912

-54 89

-16 92

80

-8214

-17 11

-87 04

-34 87

-7199

-265

-64 84

-!I 73

Table4 28

SNR

IV,

Discrepancy between !he mmimum mean-square error sequence and !he mimmum phase sequence wah 60-taps m !he filter D_

Channell

Channel2 -

Channe13

Channel4

20

-8 69

-8 93

-610

-3 08

30

-18 55

-18 81

-15 55

-1111

40

-2853

-2879

-25 46

-19 87

50

-3857

-38 82

-35 47

-2745

60

-4854

-4874

-45 08

-3799

70

-5842

-58 11

-53 19

-4727

80

-67 63

-66.15

-61 61

-5430

Table4.29

Mean-square error m the equalised Signal usmg scheme 4 with 30-taps in !he linear filter D

92

SNR

ChannelS

Channel6

Channel7

Channel 8

20

1.51

064

673

9 38

30

-7 82

-823

-078

3 61

40

-16 55

-15 86

-7 47

-3 23

50

-25 81

-2527

-16 84

-4 12

60

-3575

-34 01

-17 73

-8 63

70

-4409

-4341

-2714

-8 00

80

-5144

-53 90

-3690

-19 41

Table4.30

SNR

Mean-square error m the equahsed Signal usmg scheme 4 With 30-taps in the lmear fllterD

ChaMell

Channe12

Channe13

Channel4

20

-8 69

-8 93

-610

-3 08

30

-18 56

-18 81

-15 56

-1113

40

-2853

-2879

-25 47

-2029

50

-38 58

-38 83

-35 51

-29 43

60

-48 58

-48 81

-45 49

-38 85

70

-58 55

-58 80

-55 47

-4798

80

-6854

-6879

-6539

-57 58

Table 4.31

Mean-square error in the equaliSed Signal usmg scheme 4 with 40-taps in the hnear fllterD

SNR

ChannelS

Channel6

20

I 46

054

30

-797

-899

-I 51

3 41

40

-1724

-18 89

-977

-3 66

50

-2693

-2815

-19 00

-9 81

-

Charmcl7

CbannelS

6 54

9 38

60

-3515

-36 01

-2574

-15 17

70

-45 32

-4314

-34 53

-14 68

80

-55 40

-5444

-41 28

-2177

Table4 32

Mean-square error m the equalised Signal usmg scheme 4 With 40-taps m the lmear filter D_

93

SNR

Channel!

Channel2

Channe13

20

-8 69

-8 94

-610

-3 I

30

-18 56

-18 82

-15 56

-1113

40

-28 53

-28 79

-25 47

-2040

50

-38 58

-38 83

-35 51

-3016 -39 30

Channel4

60

-48 56

-48 81

-45 49

70

-58 55

-58 80

-55 48

-47 87

80

-68 54

-68 80

-65 54

-5710

Table4 33

Mean-square error m !he equalised Signal using scheme 4 wi!h 50-taps m !he lmear filler D.

S~'R

OtannelS

Cbannel6

Channel?

ChannelS

20

I 46

054

6 50

9 37

30

-798

-8 99

-I 83

3 41

40

-17 43

-18 97

-10 17

-3 75

50

-2713

-2909

-18 57

-11 81

60

3719

-3806

-27 49

-18 94

70

-45 53

-4445

-3644

-3002

80

-53 98

-54 21

-4123

-3145

Table4 34

S~'R

Mean-square error m !he equalised Signalusmg scheme 4 wilh 50-taps m !he lmear filter D.

Channel I

Channel2

Channcl3

-

Channel4

20

-8 69

-8 94

-611

-3 09

30

-18 56

-18 82

-15 56

-1113

40

-2854

-2879

-25 47

-20 41

50

-38 58

-38 84

-35 51

-3027

60

-48 56

-48 82

-45 49

-39 83

70

-58 55

-58 81

-55 48

-4907

80

-68 54

-68 80

-65 47

-58 65

Table4.35

Mean-square error m !he equalised Signal usmg scheme 4 with 60-taps m !he hnear filterD

94

S~"R

ChannelS

Charmel6

Ch:umcl7

ChannelS

20

I 46

054

650

9 37

30

-7 98

-8 99

-I 88

3 41

40

-17 44

-18 98

-11 25

-375

50

-2723

-2923

-2065

-11 89

60

-3728

-39 38

-28 50

-21

70

-46 89

-4924

-35 56

-28 79

80

-56 61

-5874

-43 90

-33 26

Table4 36

r:n

Mean-square error m !he equal1sed signal usmg scheme 4 wllh 60-taps m !he lmear filter D.

95

---------------------------------------------------------------------------------

CHAPTERS

ALGORITHMS FOR THE ADJUSTMENT OF THE PRE-DETECTION FILTER

5.1 INTRODUCTION Polynomials are of fundamental importance in numencal methods because many functions or systems niay be approximated by them [65-66]. The z-transform of the samples y0, Y( z )

y 1, =







,

y, IS the polynomml

-1

Yo + y 1z

+ . . . + y8 z-g

...

51

This polynomial is useful and it has several properties [65-71] given below 1-

A g-degree polynomial (Eqn. 5 1) will have g roots (zeros). These roots are the solution of the polynomial when equated to zero and they may be real or complex.

n-

If all the {y.} coefficients are real, then all complex roots will appears m complex conjugate pairs.

1ii-

When a zero is found, it can be used to reduce the degree of the polynomial by removmg It from the polynomial.

iv-

Polynomials provide a good example of the use of the mvanance principle because three different transformations can be carried out on polynomials leaving them essentially unchanged as given below [66] (1)- the transformation ofY(z) into cY(z), where c IS an appropriate constant,

(2)- the transformation of Y(z) into Y(cz),

96

-----------------------------------------------------------------

(3)- the transformation of Y(z) into z"Y(z ). Numerical methods for finding and identification of the roots (zeros) of a polynomial are often required in analysis , design or implementation of various branches of applied mathematics. It is relatively easy to find algonthms for computing the roots of polynomials With real-valued coefficients , but It is much more difficult to find algorithms which solve the more complicated problem associated with complex coefficients of high degree (greater than 20) polynomials. The problem changes from one searching for roots along the real axis or complex conjugate root pairs to one searching the complex plane for the desired roots [66]. One method to obtain an approximate value of the root is to plot the function (impulse response) and determine where it crosses the z axis. The value of z for which f(z)=O represents the root. Although graphical methods are useful for obtaining rough estimate of roots, they are limited because of their lack of prec!Slon and application. Alternative methods using numerical approach rather than graphical, have been considered m th1s chapter. The roots required for the adjustment of the pre-detection filter used m this work and descnbed m Sections 3.4.1 and 4.2-4.4 are those with absolute values greater than unity. However, most of the algonthms m the literature have been found dealmg with all the roots (inside and outside the unit circle m the z-plane) except a few algonthms which have been mentioned m references 39 and 72 which deal with the reqmred roots. This chapter presents algorithms for locating or identifymg the required roots. It also compares them. from the point of view of accuracy and complexity. The first group of algorithms identify the roots with absolute values greater than unity, whereas the second group of algorithms attempt to locate those roots.

97

5.2 ROOT-IDENTIFICATION ALGORITHMS 5.2.1 Schur Algorithm This algorithm is based on a criterion suggested by Schur [72-74] which may be used to determine whether or not a given polynormal has a root lying outside or inside a given circle in the complex z-plane. This can be achieved by a particular lmear transformation. Let

f(z)

...

=

where {a.} are the complex valued sample values ofthepolynomtal andf(O)

52

;t

0.

Also let/(z) be a polynomial whose coefficients are the complex conjugate of the reverse of those of f(z), such that

/(z)

=

a:

+ ... +

+

...

5.3

...

5.4

The particular lmear transformation that has been suggested IS

T'(f(z)]

=

a;f(z)

-

a./(z)

Th1s combination results in a sequence of polynormals m an order of decreasmg degree. The constant term ofT'(f(z)] is T'(f(O)] where

T'[f(O)]

...

5.5

This constant term is of a particular mterest because It is real-valued. Furthermore, this constant decides whether f(z) is a fit polynomial to apply the transformation g1ven by Eqn. 5.4. It also decides whether or not f(z) has a root inside the unit circle in the z-plane according to the basic theorem [71-73];

98

If for some h>O, r•[f(O)] < 0, then f(z) has at least one root mside the unit circle. If

instead, T'[f(O)] >0 for integer for which r•(f(O)]

1 $i O), r•[f(O)] < 0 or the degree of th~ reduced polynomial reaches 1, with the remaimng constant greater than zero.

This procedure IS easily understood using the flow diagram shown in Fig 5 1. Cons1der now the z-transform Y(z) of the sampled impulse response of a linear baseband channel given in Eqn. 5.1. Y(z) IS a polynomial in z-1 of g degree, and can be expressed as a function of z- 1, f(z- 1). It has g roots, k of which have a absolute values greater than unity. These k roots are those roots of f(z- 1) whose magnitudes are less than umty. The algorithm given in Eqn. 5.4 will now be applied directly on

99

----------------------------------------------------------------------

f(z- 1)> to g1ve a decision whether f(z- 1) has a root lying ms1de the unit crrcle in the

z-plane. This eventually suggests that whether Y(z) has a root lying outside the unit circle in the z-plane or not. The algorithm has been tested by computer simulation over eight different models of telephone channels referred as channels 1-8. The sampled Impulse responses of these channels are as given in Tables 2.1 and 2 2. The algorithm has also been tested over the minimum phase version of channels 1-8, pven m Tables 3.1 and 3.2, where these channels have no roots lymg outside the unit circle in the z-plane. The most Important requirement m this study IS the accurate knowledge of the roots of the z-transform of the sampled impulse response of channels 1-8 that lie outside the unit circle in the z-plane. These roots were located usmg appropriate NAG (Numerical Algorithm Group) routme algorithm [60] runmngon a Honeywell DPS-8 computer; these roots along with their absolute values, are g1ven in Tables 4.1 and 4.2, respectively for channels 1-4 and 5-8. Throughout the tests the number of real arithmetic operations involved in the algorithm to give the required decision were calculated and recorded as shown in Tables 5.1-5.8. Furthermore, from the tests, the following interesting points have been found. 1-

The algonthm did not failed with any of the channels tested. The number of arithmetic operations required for each channel to give the right decision are as shown in Table 5.1 and 5.2, respectively, for channels 1-4 and 5-8.

u-

The value of T"[Y(O)] decreases ash increases, and after a certain value of h ,11 becomes very small (approx1mately zero). This major problem appeared because of the transformation given by Eqn 5.4, where at each step Y(z) or its successive modified version, has to be multiplied by y0 and y8 , bearing in mmd that these two components are less than unity This problem has been solved by checking at each step of transformation T"[Y(O)]. Whenever this component becomes less than 10-5 , all the coefficients ofT"[Y(z)] should be multiplied by 105• Th1s of course has no effect on the decision considered by the algorithm.

100

ii1-

Dropping the lower-degree components ofY(z) has no effect on the decision considered by the algonthm. Furthermore, it reduces the number of arithmetic operations involved to give the decision especially 1f the channel is minimum phase as shown m Tables 5.3-5 8.

1v-

All the minimum phase channels tested produce the same results in terms of the number of arithmetic operations involved when the number of the components in the sampled impulse response of the channel are the same. Therefore, the results of one minimum phase channel are considered here to represent all the minimum phase channels as shown in Table 5.8, where the number of components in the sampled impulse response of the channel1s the only factor which affects the number of mthmetic operations involved.

5.2.2 Nyquist Criterion The Nyqmst criterion is an analysis tool for determmmg whether or not an impulse response is mimmum phase As mentioned earlier, the impulse response of the channel is minimum phase only 1f all the roots (zeros) of the z-transform of the impulse response he ms1de the umt crrcle m the z-plane Therefore, a contour which encloses the entrre roots that lie outside the circle in the z-plane must be chosen, determimng whether any of those roots lie withm that contour. This can be achieved by utilizing complex variable theorem known as the argument pnnciple [75-80]. The Nyquist criterion follows directly from an important theorem of complex variable theorem, which states: If f(z) is analytic and dtfferent from zero on the contour C, then [75-80]

5.6

where N IS the number of zeros of f(z) lymg inside the chosen contour C and f (z) is the derivative of f(z). Furthermore, since f(z)

101

IS

analytic and further since

--------------------------------------------------------------------------

/(z) f(z)

=

d -- [lnf(z)]

...

dz

5.7

Eqn. 5.6 can be wntten as

f~&~ dz =

L'.Jlnf(z)]

=

j2rr.N

000

5.8

But L'.Jlnf(z )]

=

+ j L'.Jargf(z )]

L'.Jlnlf(z )I]

000

59

where L'.Jx] denotes the vanation in x around the contour C. Therefore, it can be readily be shown that

f~g~

dz

=

j L'.Jarg [f(z )]]

000

5.10

since C is closed contour, lnlf(z )I cannot change around it and therefore this term is zero [75-80]. The proof ofEqn. 5.10 together with further details about the theorem can be found elsewhere [75-801 . The argument principle can also be interpreted geometrically. Further details on this mterpretation are given elsewhere [76]. Consider now the z-transform of the sampled Impulse response of a lmear baseband channel, Y(z), given in Eqn 5.1. Y(z) is as defined earlier, a polynomial of g degree m z -I and can be expressed as a function of z-1,f(z- 1). Forthe application considered here, It IS necessary to identify the roots of f(z- 1) for which the magnitudes of z is greater than unity. The root-i~~UfJcauon algorithm suggested in Eqns. 5 6 and 5.10 is applied directly to [(z-1)'fg1ve the value of k (the number ofroots ofY(z) that lie outside the unit circle). The roots of[(z-1) whosemagnitudes are less than unity are, therefore, the roots of Y(z) which he outside the unit circle. The contour proposed here is taken to be the unit Circle m the z-plane.

102

Before applying the algonthm, Y(z) in Eqn. 5 1 is modified such that each z is replaced by e16 to give Y(S), where

...

5 11

...

5.12

Since

hence, Eqn. 5.11 becomes Y(6)= Yo + y 1(cos6-Jsm6) + =

Re(6)

. . + y,(cosge- 1 smg6)

+ lm(6)

513

where Re() and Im() are the real and the imaginary part ofY(S), respectively. From Eqn. 5.13, the argument of Y(S) can be calculated, such that;

=

arg[Y(S)]

The value

of~

~

=

tan-{~:~:n

...

5.14

in Eqn. 5.14 has an important function m this algonthm. It decides

how many roots he inside the umt circle (contour) when 8 vanes from 0- 21t Here, the number of roots equals the number of times the value of~ exceeds 21t. The Nyqmst cntenon descnbed above, has been Simulated by computer as follows: i-

Let m represent the number of roots Jymg inside the unit circle. This value initially has been set to zero together With the value of8 in Eqns. 5.11-5.13.

n-

The value of Y(S) as suggested in Eqn. 5.13 can be determined. Having determmed Re(S) and Im(S), the value Eqn. 5.14.

103

of~

can also be determmed using

m-

The value

of~

IS checked, and whenever

~

exceeds 27t, the value of m is

incremented by unity. IV-

The value of9 will be incremented by t-9, and steps1i and iii will be repeated, until 9 reaches 27t. If 9 = 27t, the algorithm will terminate and the value of m will be equal to k, where k represents the number of roots lying inside the unit circle.

Fig. 5.2 shows the flow diagram of the above procedure, which has been Simulated by computer program as shown in the appendices. The Nyquist cntenon has been tested by computer simulation over different models of telephone channels. The sampled impulse responses of these channels are g1ven in Tables 2.1 and 2.2. Throughout the tests the number of arithmetic operatiOns involved to decide how many roots he outside the unit circle have been calculated and recorded as shown in Tables 5.9-5.17. It is clear from these tables that the mam factors which affect the number of operatiOns aret.9 and the number of components in the sampled impulse response of the baseband channel (the value of g in Eqn 5.11 ). When the decision is correct, It has been found that the level of distortion in the channel has no effect on the number of arithmetic operations involved. However, channels mtroducing high level of d!stortion require a small value of t.9 and long impulse response to give the correct decision. Therefore, the results in Tables 5.95.17 represent all the channels tested, since all the channels give the correct decisiOn. It has been recogmzed that as the value of t.9 mcreases, the number of arithmeuc operatiOns mvolved w1ll decrease until .at a certam value of t.9 the algonthm fails to identify all the k roots inside the unit circle. Furthermore, reducmg the value of g has the same effect as mcreasing the value of t-9. Tests over channels 1-4 show that when the value of t.9 IS increased up to 20 degrees and g reduced to 11, the algonthm can still Identify the correct number of k roots for each of the channels tested. Throughout the tests the number of real arithmetic operations involved were calculated (addiUon, subtraction and multiplication), and recorded as shown in Tables 5.9-5.17. Furthermore, tests over channels 1-3, whose results are not recorded, have shown that when the value of g is reduced to 8 and t.9 mcreased up to 24 degrees the algorithm still can correctly Identify the exact number of roots.

104

Tests over channels 5, 6 and 8 have shown that the algorithm can correctly identify the roots with the maximum value for Ll9 set up to 14 degrees and minimum value for gas 19. The results of these tests which show the number of arithmetic operations required are exactly the same as those recorded in Table 5.13. Tests over channel 7 show that this channel reqmres more operations as compared with the other channels tested, since the minimum value of g and the maximum values of Ll9 are, respectively, 41 and 6 m order to identify the exact number of roots as shown in Table 5.14. The algorithm has also been tested over the minimum phase versions of channels 1-8 These channels should give a value ofkequal to zero. Therefore, the algonthm should give the same results over any minimum phase channel. It has been found that the variation of



as 9 increases from 0- 27t is entirely d1fferent from

non-m1mmum phase channels. In the latter channels vanes stead!ly as 9 increases whereas with the former channels



oscillate around zero. This phenomenon has

been considered m the termination of the algonthm before 9 approaches the value of 21t. The results of this algorithm over the minimum phase channels are as shown m Table 5.15-5.17, for d1fferent values of g. Fmally, the algonthm involves the determinatiOn of trigonometric equations (sin, cos, arctan); these functions for different values of 9 can be stored in a look-up table and can be called through the execution of the algonthm. Therefore, the results presented m Tables 5 9-5.17 have ignored the evaluation of these functions m the cost calculation. 5.3 ROOT-FINDING ALGORITHMS 5.3.1 Algorithm 1 Here, the general form of Newton's method (also called Newton-Raphson iteratiOn) is by far the most popular method in numencal analys1s for locatmg the roots (zeros) of polynomial equations [67]. This method approximates the roots through an iterative process. It IS extended to cope with complex-valued polynomial, f(z). Fig. 5.3 shows the geometric mterpretation of Newton's method for the simple case of a real root.

105

---------------------------------------------------------------------------------

The first step in th1s method

IS

to make the initial guess for the m'• root; this is

abbreviated as 'Ym,o· The lme tangent to f(z) at the pomt C'Ym,oJC'Ym,o))

IS

next

determined. The intersection of this line with the z axis is the point 'Ym,I· As seen by the geometric interpretation, 'Ym,I represents a better es!lmate of the root of f(z). By repeating this process, a much better estimate w11l be made. From Fig 5 3,

=

tanS

fC'Ym,o) 'Ym,! -Ym,o

where/C'Ym 0 ) denotes the derivative off(z) at z

=

'Ym,l

'Ym,O

= "fm

0•

-

. .

5.15

...

5.16

...

5.17

Hence

In terms of an iterauve algonthm,

'Ym,l + 1

=

'Ym,a

-

Apart from the geometnc denva!lon, Eqns. 5.15-5.17, may also be developed from the Taylor senes expansion. This alternallve derivation

IS

useful in that it provides

an insight into the rate of convergence of the method [65-70]. Let !:J.z be a suitable value such that

J("fm

0

+ !:J.z) = 0

...

5.18

Expandmg the above equatiOn by Taylor series gives

f("fm,O

+ .:'.z)

+ ...

106

5 19

where /(z) denotes the second derivative of f(z). When Ym,o becomes close to the value of the root the approximate version to the above equation is obtainable by truncating the series after the flrst denvative where the factors associated with !lz become very small. Furthermore, at the value of a rootf(Y,.,o + ru) =0. Therefore, From Eqn. 5.19, the value of !lz is giVen by -J(Y,.,o)

/(Y,.,o)

...

5.20

...

5 21

which can be solved for

'Ym,1+l

=

J(y,.,,)

'Ym,l

-

--

/(y,...)

which is identical to Eqn 5.17. The Iterative process defined in Eqn 5.21 is repeated unull Ym,•+t -y,.,.l < d, where d IS some small, real-valued quantity. When this is obtained, the process seems to have converged to a root of the polynomial, within the accuracy set by the value of d. An alternative stopping cri tenon has been also used to test if lf(y,...)l

>

f

.-

0

0 0

~

_,.

0 0

_.,

0

0 0

Legend


-

T

r

X

~·I

. r

fl,-1' fi,O' f l , l ' ' ' " ' f I ,g

1:

Fig. 5.40 Two-tap feedforward transversal filter

.

"

. ... .-. ... . '\ ... .... \ \ _,.

~

~

~

~

~

~

. ~

\

~

•:;- _,.

0

\

~

\ : ... I \ I _, \ _, .. f

. .. •: •.. ... . ... •. .: ... ... ... ~

_,.

Legend

~

'

Root 1

~Root 3



_,.

_,.

~



Number or llorollon (I)

. .

-..

.

• _,. ~

... ~

~

•. _, ~

..

. .; _,-··

~

~

. _,. •. _, . _., 0

0

~

• ...

~

Legend Root 2

_,.

... ...

\

1\ \\ I

I

I I

\

\

\

I

Legend Root 1

\ \

\

~~oiL

•Humber olltoraflon " (I)

..

Root 4

"

.

...

I

flg, 5 •.43 Performance of algorithm 5 with channe13

rtg. 5.41 Performance of algorithm 5 with channel1

..E •

('\

f

:

Root 1 _

Root

~

L

... ...

Root 4

~

\

.



.

Num!Jor of ll.rotlon (I)

"

Fig. 5 42 Performance of algorithm 5 with channel 2

''

-''

'' '' '' '' '' '' ''

'' ''

\

Legend Root 5

'

\ ~I

'' '' '' '' '' '' '' '' 1 ' " ollluollon (I) Humbor

\I

~Root

L

Root 1

B.2.2!LR~o!~--

___

8~!'Ls

Root 2

..

Fig. 5.44 Performance of algorithm 5 WLfh channe14

142

__ _ _ _ j

" _,

.. -··

~

Legend

.... .•

= •

a....J..L_.

Btlll.~t!_

Legend -11

llil..L_

!U1,!_ J!u.U_

Wll..-

!.•.!f.J L -

!ll!..!...._ !l.uJ_L_

!!!.!.!..!.!...

!.•.!U_-

!!.!1..!..!!.!!..ll...-

!!.!.1.]__

!tl''_., _.,

!~1!1_7_

ltuJ..!_

__

_.,

b.!!.L_,

Fig. 5.45 Pnformanct of algorithm 5 with channtl5

rig. 5.47 Perform a ne• of algorithm 5 with chonntl7

.

"

·-• . -·· : ,;

,;

•: .... :

-.

;;"'

••

-··

. -·· =

!.!.!!..!_.

!ll!...!..!!.!14_

ii -n

!.tlll_



a....J..L_. Root 11

;

. ...

!.•.!'-'- _

;:

!.!.!!..1L IW!..1L

~

f

;

!t.'l•-•---

•71

_.,

!.t!!..!..._

~ Root 10 _

';

!!.!!..!..._ !.•.!1_3_-

Legend

f

:. _,.

Legend

!\!.tll_

_.,

~ll..-

!.!..!!...L_

!t.!.I.J•••

ll.Ui.!.~

IWU..~--

_,.

_.,

Fig. 5.46 Performance of algorithmS with channel 6

Fig. 5.48 Perform one• of algorithm 5 with chann•l B

143

----------------------------------------------------------------------------------------

Operatton

Channell

Channel2

Channel3

Channe14

AddJ.uon&

393

393

317

483

560

548

468

738

Subtraction

Mult1pltcauon

TableS 1

Number of anlhmeuc operauons mvolved over channels 1-4 to g1ve the decu10n, when applymg

Schur algonthm

Table 52

Operat100

ChannelS

Channel6

Channel?

ChannelS

AddtUon& Sublracuon

3

193

3

3

MuluphcatJon

4

256

4

4

Number of anthmeuc operauons mvolved over channeb 5-8 to glVe the dectSIOO, when applymg

Schur algonthm

Table53

TableS 4

Table 55

Operat100

Channel I

Ol.anne12

Channel3

Channe14

AdditlOD & Subtracuon

463

357

245

127

Multtphcat!On

680

542

360

168

Number of anthmettc operauons mvolved over channels 1-4 to g1ve the declSlon, when applymg Schur algonthm With g set to 19

Operauon

Channel!

Channe12

Channe13

Channe14

Add1Uon& Subtract10n

343

97

97

97

Multipltcatton

500

128

128

128

Number of anthmettc operations mvolved over channels 1-4 to gtve the declS!On, when applymg Schur algonthm With g set to 14

Operauon

Channel I

Channel2

Channcl3

Channel4

Add111on & SubtractiOn

67

3

3

3

Mult1phcauon

88

4

4

4

Number of anthmel!C operaUons mvolved over channels 1-4 to gtve the dcctston, when applymg Schur algonthm w1th g set to 9.

144

Operatlon

Channel I

Channel2

ChaMel3

Channe14

Addtuon & Subtracuon

3

3

3

3

Muluphcauon

4

4

4

4

T•b1e 56

Number of anthmeuc operauons mvolved over channels 1·4 to gtve the dectston, when applymg Schur algonthm Wtlh g set to 5

Operatton

ChannelS

Channel6

Chatmel7

ChannelS

AddtUOn & Subtracuon

3

3

3

3

Mulupltcauon

4

4

4

4

Number of anthmettc operauons mvo]ved over channels 5·8 to gtve the dectston, when applymg

T•b1e 57

Schur algonthm Wtlh g set to the values between 4 and 25

Operatton Addttton& Subtracuon Multtpltcatton

Table 58

g=4 100

g=9 360

g=14 770

g=19 1330

g=24

g=29

1903

2900

128

468

1008

1748

2524

3850

Number of anthmeuc opcrauons mvolved over the mm unum phase verSion of channels 1·8 to gtve the dectston, when applymg Schur algonthm wtth dlfferent values for g

Operauon

4

Addtuon& SubtractiOn

5824

6 3904

8 3008

MulttphcattOn

5005 91

3355 61

2585

Dtv!Sion

Table 59

47 -

10

12

14

20

1989

1728

16 1536

18

2368

1349

1280

2035 37

1705

1485 27

1320 24

1155 21

liDO

31

20

Number of anthmettc operations mvolved over channels 1-4, for dUferent values of 6.9 Wtlh g set to 11

Operatton Addmon & Subtractton Multtphcatton Dtvtston

Table 510

4 6734

6 4514

8 3478

10

12

14

16

18

2738

2294

1998

1776

1554

20 1480

5915 91

3965 61

3055 47

2405 37

2015 31

1755 27

1560 24

1365 21

1300 20

Number of anthmcttc operattons mvolved over channels 1·4, for dlfferent values of 6.9 wtth g set to 13

145

----------------------------------------------------------------------------------------------------

Operauon

4

6

8

10

12

14

Addtuon & Subtracuon

7644

5124

3948

3108

2604

2268

16 2016

18 1764

20 1680

Muluphcauon

6825

4575

3525

2775

2325

2025

1800

1575

1500

DtVlSlOR

91

61

47

37

31

27

24

21

20

Table 5 11

Number of anthmeuc operations mvolved over channels l-4, for dtfferent values of ~9

v.1th g set to 15

Operatton Addmon &

4 8554

6 5734

8

10

12

14

16

18

20

4418

3474

2914

2538

2256

1974

1880

7735 91

5185 61

3995 47

3145 37

2635

2295 27

2040 24

1785 21

1700 20

SubtractiOn

Mult1phcatton DtVISlOO

T•ble 512

31

Number of anthmeuc operauons mvolved over channels 1-4, for different values of ~9 wtth g set to 17

Operattan

4

6

8

10

12

14

16

18

20

Addmon & SubtractlOn

9464

6344

4888

3848

3224

2808

2496

2184

2080

Multtphcauon

8645

5795

4465

3515

2945

2565

2280

1995

1900

Dtvtston

91

61

47

37

31

27

24

21

20

Table513

Number of anthmeuc operattons mvolved over channels 1-4, for dtfferent values of .19 wtth g set to 19

Operat10n

t.9=4,g=4L

69=4,g=39

69=6, g = 41

Add!tton & Subtractton

19474

18564

13054

Mult•plicatton

18655

17745

12505

DIVlSlOO

91

91

61

TableS 14

Number of anthmettc operations mvolved over channel?, for dtfferent values of .19 and g

146

Operation

4

8

12

16

20

24

Addttton & Subtractton

2211

1131

807

591

483

429

28 375

32 321

Mu1ttpltcatton

1845

945

675

495

405

360

315

270

Dtvtston

41

21

15

11

9

8

7

6

Table 515

Number of anthmettc operauons mvolved over the mm unum phase verSion of channels 1-8, for dtfferent values of .6.9 wtth g set to 9

4 3315

8 1735

12 1182

16 945

20

24

787

629

28 580

550

Mu1ttphcauon

2940

1540

1050

840

700

560

490

440

Dtvtston

42

22

15

12

10

8

7

7

Operatton Addttton& Subtracuon

Table 516

Number of anthmetlc operauons mvolved over the mm unum phase verSton of channels 1-8, for dtfferent values of .6.9 wtth g set to 14

Operation

4

8

12

Addttton& Subtractton

4365

2285

1557

16 1245

Multtplicatton

3990 42

2090 22

1425 15

1140 12

Dtvtston

Table 5 17

20

24

28

32

1037

829

725

725

950

760 8

665

665 7

10

7

Number of anthrnettc operattons mvolved over the mmunum phase verston of channels 1-8, for different values of .6.9 wtth g set to 19

Operatton

Channel!

Charmel2

Channel3

Channe14

Addtuon& Subtracuon

34109-

27607

39846

44618

Mu1upl.tcauon

29874

24160 448

34932

39122

566

626

Dtvtston

Table 5 18

32

586

Number of anthmeuc operattons mvolvcd when applymg algonthm 1 over channels 1-4, wtth d set to 1 x 10_.

147

---------------------------------- -----------

Root number

Channel!

Channel2

Channe13

Ch.annel4

Root 1

-131 0

-134 7

-168 9

-141 I

Root2

-127 6

-135 4

-1712

-141 2

Root3

-1346

-130 2

-142 5

-139 3

Root4

--

-140 7

-1461

-138 8

----

---------

-144 8

RootS Root6 Root7

~~--

RootS

----

---

-222 9 -1325 -1460

Roots accuracy obtarned usmg algonthm 1 over channels 1-4 wtth d set to 1 x 10.....

Table 519

Oper.mon

ChannelS

Channe16

Channel?

ChannelS

Addtuon& Subtracuon

64313

47536

68654

37066

Muluphcat10n

56506

41692

60244

32458

620

620

538

746

DtVlSIOD

Table 5 20

Number of anthmettc operauons mvolved when applymg algonthm 1 over channels 5-8, With d set to

Root number

tcr OJ.annel5

Charmel6

Charmcl7

Root 1

-92.40

-16193

-178 17

-97 40

Root2

-200 56

-65 85

-139 49

-113 83

Root3

-87 46

-67 97

-134 67

-173 04

Root4

-97 48

-154 56

-174 11

-133 68

RootS

-188 29

-7014

-165 35

-110 87

Root6

-94 97

-66 43

---

-120 05

Root?

-190 17

-74 80

-119 60

-102 63

RootS

-9674

-158 35

-114 43

-115 47

Root9

-8290

Root 12

------------

Root 13

--

Root 10 Root 11

TableS 21

1X

----------

ChannelS

-

-122 65

------

-115 01

Roots accuracy obtamed usmg algonthm 1 over channels 5-8 wtth d set to 1 x 10_.

148

-114 79

-171 90

-131 84 -121 75

OperatiOn

Channel I

Channe12

Channe13

Charmel4

Addtuon & SubtracUon

34973

28776

52890

46798

Muluphcauon

30632

25188

46410

41042

468

766

658

DtVlSlOR

Table522

584

Number of anthmeuc operauons mvolved when applymg algonthm 1 over channels l-4, wtth d set to 1 x 10""'

Root number

Channel!

Channel2

Channel3

Channe14

Root 1

-25077

-257 22

-32193

-23142

Root2

-238 20

-24161

-275 94

-228 54

Root3

-237 60

-244

so

-222 09

-221 81

Root4

-20000

-22263

RootS

------

-153 64

--

Root?

--

--------

---------

-220 89

Root6 RootS

Table 5 23

-222 90 -228 58 -230 62

Roots accuracy obtamed usmg algonthm 1 over channels 1-4 wtth d set to 1 x to""'

Operation

Channel 5

Channel6

Channel?

ChannelS

Addttton & Subtractton

67926

51301

59154

39935

Mult1phcauon

59692

45012

52394

34986

668

510

578

DtVISIOR

Table 5 24

792

Number of anthmeuc operations mvolved when applymg algonthm 1 over channels 5-8, Wtth d set to 1 x la""'

-

149

Root number

ChannelS

Channel6

Channel?

ChannelS

Root 1

-140 52

-313 04

-342 09

-156 58

Root2

-219 54

-212 73

-228 80

-210 53

Root3

-138 52

-215 06

-229 74

-338 49

Root4

-147 86

-294 71

-231 53

-249 30

RootS

-213 51

-217 22

-225 10

-208 21

Root6

-140 39

-213 50

----

-217 85

Root?

-356 94

-221 47

-223 98

-159 68

RootS

-143 43

-297 79

-22023

-214 87

Root9

-------

-229 56

---

-180 51

------

-309 48

-219 97

--

-176 69

-----

-24027

Root 10 Root 11 Root 12 Root 13

---

-222 68

Roots accuracy obtamed usmg algonlhm 1 over channels S-S With d set to 1 x 10~

TableS 25

Ope rat ton

Channel I

Channe12

Channc13

Ch.anne14

Addiuon &. Subtracuon

34565

28412

43731

46330

Mulliphcatton

30296

24890

38382

40656

DIVISion

536

422

562

604

Number of anlhmet1c operations mvolved when applymg a]gonlhm 1 over channels 1-4, w1th ~ set to 1 x lOo~;

Table 5 26

Root number

Channel I

Rootl

-131 0

Root2

-118 5

Root3

-121 2

-126 4

-91 9

-123 3

Root4

---

-130 5

-962

-114 2

-----· -----

-140 4

----

------

RootS Root6 Root7 RootS

TableS 27

-

--

Otannel2

ChaMe13

Cltannel4

-135 4

-92 2

-120 I

-124 5

-94 2

-126 0

Roots accuracy obtamed usmg algonthm 1 over channels 1-4 wtth ~set to 1x

150

-119 8 -1157 -109 8

tcr

Operatton

Channel5

Channel6

Channel7

ChannelS

AddJ.uon& SubtractiOn

67444

50726

57771

37127

Mulupl..!catwn

59298

44532

50658

32550

612

462

470

Dt.,.,ston

730

Number of anthmettc operauons mvolved when applymg algontltm 1 over channels S·S, wtth ~set to 1 x 10-4

TableS 2S

Root number

ChannelS

Channel6

Channel7

ChannelS

RootL

-lOO 24

-86 37

-9618

-79 82

Root2

-109 66

-103 38

-9259

-90 01

Root3

-112 95

-10167

-99 06

-87 82

Root4

-11441

-8267

-9456

-7629

RootS

-102 22

-9442

-90 33

-86 96

Root6

-106 84

-84 39

··-·

-75 13

Root7

-105 99

-91 00

-98 19

-83 55

RootS

-107 33

-87 48

-93 25

-83 61

Root9

----------

-9055

---

-8400

------

-97 28

-7664

···-

-84 83

-----

-82 85

Root 10 Root 11 Root 12

Rootl3

Table 5 29

-85 28

Roots accuracy obtamed usmg algonthm 1 over channels S-8 With~ set to 1 X I er'

Operatton

Channel I

Channel2

Channel3

Channel4

Addttion& Subtractton

35761

29920

53789

48439

Multtpl..!catton

31348

26216

47228

42514

448

732

634

DtVlSlOn

Table 5 30

-

556

Number of anthmettc operanons mvolved when applymg d.lgonthm 1 over channels 1-4, With ~ set to 1 x 10_..

151

Root number

Channel!

Root 1

-250 8

-2572

-168 9

-141 I

Root2

-238 2

-241 0

-1406

-141 I

Root3

-169 2

-2041

-144 6

-139 3

Root4

---

-2104

-146 0

-138 8

RootS

-

----

--

-144 8

-----

-2229

Root6

---

Root?

-----

RootS

Channel2

....

--

Channe13

Channel4

-146 0 -146 0

Roots accuracy obtamed usmg algonthm 1 over channels 1-4 wllh l; set to 1 x 10.....

Table 5 31

Operation

ChannelS

Channel6

Channel?

ChannelS

Addtuon& Subtractton

69796

51975

60832

40897

Multlphcallon

61372

45630

53360

35860

634

496

544

DtV!StOR

Table 5 32

760

Number of anthmettc operauons mvolved when applymg algonthm 1 over channels 5-8, wuh l; set to 1 x 10_.

Root number

ChannelS

Channel6

Channel?

ChannelS

Root 1

-164.56

-16193

-178 17

-134 88

Root2

-200 55

-14714

-171 82

-135 67

Root3

-169 4

-145 48

-16168

-173 04

Root4

-171 85

-154 56

-17411

-133 68

RootS

-188 29

-147 76

-165 35

-132 22

Root6

-168 83

-144 60

---

-128 55

Root?

-19017

-154 09

-17213

-132 89

RootS

-159 6

-158 35

-158 76

-137 22

Root9

-------------

-155 35

-129 05

------

---171 90

-126 80

----

-138 07

--

-131 84

----

----

-139 79

Root 10 Root 11 Root 12 Root 13

Table 5 33

-

Roots accuracy obtamed usmg algonthm 1 over channels 5-8 wtth l; set to 1 x 10_.

152

----------------------------------------------------------------------------------------------

Operation

Channel I

Channel2

O.anne13

Channel4

Addtuon&

40943

35147

59160

54043

35932

30840

51980

47472

488

772

676

Subtntcuon Mulupltcauon Dtvtston

596

Number of anthmettc operattons mvolved when applymg extended algonthm 1 over channels 1-4

TableS 34

Root nmnber

Channel!

Channel2

Channel3

Channel4

Root 1

-131 0

-253 65

-168 89

-30192

Root2

-248 66

-254 86

-208 45

-267 21

Root3

-245 95

-135 43

-272 85

-371 34

Root4

---------

-272 89

-316 26

-364 35

-----

--

-363 73

--

-2229

--

---

-244 92

RootS Root6 Root? RootS

----

-362 06

Roots accuracy obta.med usmg extended algonthrn 1 over channels 1-4

Table 5 35

Operattan

ChannelS

Otannel6

Channel?

ChannelS

Addtuon& Subrracuon

75371

58064

343837

46598

Multtphcauon

66302

51018

304080

40902

678

2008

590

DtV!SlOR

Tab-le 5 36

806

Number of anthmeuc operauons mvolved when applymg extended algonthm 1 over channels 5-8

Root number

ChatmelS

Channel6

Channel?

ChannelS

Root 1

-276 00

-16193

-178 17

-287 47

Root2

-365 91

-300 05

-259 67

-273 67

Root3

-312 80

-27719

-306 90

-173 04

Root4

-305 43

-320 30

-325 75

-248 0

RootS

-352 01

-336 82

-32714

-330 00

Root6

-171 06

-315 94

---

-216 16

Root?

-190 17

-278 31

-310 91

-32111

RootS

-268 85

-293 24

-309 78

-313.52

Root9

------

-263 94

-31167

-321 35

Root 10 Root ll

Root 12 Root 13

TableS 37

-

-----

----

-310 06

-262.23

--

--

-333 54

------

-----

-239 87

Roots accuracy ob-tamed usmg extended algonthm 1 over channels S-8

153

-295 30

Oper:lliOn

OlaMell

Channel2

OlaMel2

Olannel4

Addtuon& Subtracuon

3262

2125

3680

10923

Muluphcauon

2802

4428

3168

9494

DIVISIOn

118

172

128

270

Table 5 38

Number of anthmettc operauons mvolved when applymg algonthm 1 over truncated channels 1-4

Rootnwnber

Channel!

Channel2

ChaMel3

Channel4

Root 1

-1309

-156 9

·122 3

-132 3

Root2

-104 8

-121 0

-61 5

-144 1

Root3

-641

-156 9

-502

-103 6

Root4

·-·

-38 2

-264

-760

RootS

----

-··

-655

Root6

-



Root7 RootS

·--·

----

----·--

-··

-440 -270 -240

Roots accuracy obtamed usmg a1gonthm 1 over truncated channels 1-4

Table 5 39

OperatiOn

ChannelS

Olannel6

Olatmel7

CbaMelS

Addttton & Subtracuon

20891

18195

19733

17316

Multtphcatton

18240

15868

17224

15110

352

372

310

DtvlSIOn

TableS 40

396

Number of anthmettc operauons mvolved when applymg algonthm 1 over truncated channels 5-8

Rootnwnber

ChannelS

OJ.annel6

OJ.annel7

Olannel8

Root 1

-8625

-161 92

-178 17

-13021

Root2

-7636

-141 80

-128 45

-142 36

Root3

-134 78

-14165

-92 52

-10014

Root4

-7743

-7973

-6523

-99 78

RootS

-6000

-8374

-6015

-9707

Root6

-48 97

-44 41

-30 94

-41 70

Root7

-2426

-25 12

-18

so

-6200

RootS

-2002

-16 7

-23 35

-74

Root9

--

-21 0

----

-22 68

Root 10

----

Root 11

-··

·-· ·-·

Rootl2

-· --

Root 13

Table 5 41

.

-··

-8 31

-

----

-30 00

·-·

--·

-65

Roots accuracy obtamed usmg a]gonthm 1 over truncated channels 5-8

154

-16 9

--------------------------------------------------------------------------------------------------------------,

Operation

ChaMell

Channel2

Channel3

Cbannel4

Add1Uon & Subtracuon

7352

12211

18038

19683

Muluphcatton

6320

10532

15602

17148

Dtvlston

140

216

292

222

Table 5 42

Number of anlhmeuc operations mvolved when applymg algonthm 2 over channels 1-4

Rootnwnber

Channel I

Cbannel2

Channel3

Channe13

Rootl

-131 0

-134 7

-168 9

-1411

Root2

-127 6

-135 4

-144 6

-141 2

Root3

-1206

-1302

-110 3

-1393

Root4

--------

-118 0

-121 0

-1372

---

-122 3

--

-------

-

--

-742

RootS Root6 Root? RootS

TableS 43

-

--

-94 0 -75 I

Roots accuracy obtamed usmg a1gonthm 2 over channels 1-4

Operat1on

ChannelS

Channel6

Channel?

ChannelS

Addmon &

60165

62402

75599

69485

51936

53562

65604

59636

988

1176

994

1350

SubtracUon

Mu1t1pbcauon Dlv!ston

Table 5 44

Number of anthmellc operauoos mvolved when applymg algonthm 2 over channels 5-8

Rootnwnber

ChannelS

Channe16

Channel?

ChannelS

Root 1

-188 94

-16193

-178 17

-11791

Root2

-200 56

-174 67

-139 49

-113 83

TableS 45

-

Root3

-189 62

-128 24

-134 67

-173 04

Root4

-187 80

-154 56

-174 11

-133 68

RootS

-188 29

-12750

-165 35

-110 88

Root6

-187 09

-125 ()()

-14617

-12005

Root?

-19017

-132 94

----

-111 53

RootS

-119 21

-158 35

-104 83

-115 47

Root9

-142 70

-12033

-121 34

Root 10

-----

---

-17190

-122 65

Root 11

--

----

----

-118 11

Rootl2

----

----

Root 13

----

----

Roots accuracy obtamed usmg algonthm 2 over channels S-8

155

------

-131 84 -121 75

Operatton

Channel!

Channel2

Channel3

Channel4

Addttton & Subtraction

5733

8783

11070

24977

Multtpltcat!on

4642

7132

8962

20252

DtvlSton

54

84

90

204

Number of anthmeuc operat.J.ons mvolved when applymg algonthm 3 over channels 1-4

TableS 46

Root number

Channel!

Channe12

Chatmel3

Chatmel4

Root 1

-131 0

-1119

-168 9

-108 I

Root2

-127 6

-135 4

-110 9

-107 7

Root3

-113 8

-107 3

-106 9

-1127

Root4

-1122

-lOO I

-103 8

---

Root7

----

RootS

---

------

---------

-131 I

Root6

-------

RootS

Table 5 47

--

-912 -668 -59 5

Roots accuracy obt:amed usmg algonthm 3 over channels 1-4

Operatton

ChannelS

Channel6

Channel?

ChannelS

AddttJOn & Subtraction

20912

17686

44677

23441

Multtpltcat!On

16882

14284

35986

19008

136

234

208

Dtv!Ston TableS 48

Kumberof anthmeuc operattons mvolved when applymg atgonthm 3 over channels 5-8

Root number

ChannelS

Channe16

Channel?

ChannelS

Root 1

-10434

Root2

-120 81

-161 93

-178 17

-138 59

-107 53

-10078

Root3

-114 40

-108 47

-106 49

-10158

Root4

-173 04

-112 93

-120 91

-102 96

-99 66

RootS

-106 82

-115 28

-96 68

-93 66

Root6

-91 00

-9626

---

-74 26

Root7

-47 97

----

-7268

-84 72

-53 35

-68 21

RootS

-

-

----

Root9

---

--

---

Root 10

--

-46 84

-39 34

Rootll

--------

------

---

Root 12 Root 13 Table 5 49

ISO

Roots accuracy obtamcd usutg a]gonlhm 3 over channels 5-8

156

-------

-5034 -3678

---

Operauon

Chatu1ell

Chatulel2

Channel3

Channel4

Addruon & Subtraction

12315

14650

29529

376:!6

Multtpltcauon

10566

12676

25830

32944

248

340

410

DtvtSlOn Table 5 50

260

Number of anthmeuc operauons mvolved when applymg algonthm 4 over channels 14

Root number

Channel1

Chatu1el2

ChaMel3

Channel4

Root 1

-131 00

-134 67

-168 89

-163 62

Root2

-127 61

-13543

-168 45

-51 38

Root3

-134 73

-13019

-168 99

-48 14

Root4

···-

-139 28

-172 01

-43 89

------

---

RootS

---

Root6

----

Root?

----

RootS

---

-46 01 -222 90

---

-45 24 -45 82

Roots accuracy obtamed usmg algonthm 4 over channels 1-4

Table 5 51

Operauon

Chatu1el5

Channel6

Chatu1el?

ChannelS

AddtllOn & Subtracuon

41245

50089

113452

54061

Mult1pltcauon

36206

43870

100086

47444

534

684

528

DrviSlon Table 5 52

380

Number of anthmeuc operauons mvolved when applymg algonthm 4 over channels 5-8

Rootnwnber

Chatu1elS

Channel6

Channel?

Root 1

-168 42

-16193

-17817

-9094

Root2

-187 66

-102 91

-147 99

-112 21 -173 04

Root3

-78 65

-10152

-68 33

Root4

-174 97

-144 49

-70 35

-87 62

RootS

-67 90

-9919

-128 22

-11116

Root6

-166 66

-9725

-73 64

-104 26

Root7

-190 17

-149 97

-160 88

-9145

RootS

-150 49

-172 62

-139 09

-8029

Root9

--------··----

-98 50

-138 06

-140 26

-··---

-189 15

-102 98

-179 90

-8630

----

-142 86

-80 89

----

---

-8513

Root 10 Root 11 Root 12 Root 13 Table 5 53

ChannelS

Roots accuracy obtamcd usmg algonthm 4 over channels 5-8

157

Operatton

Channel I

Channe12

Channel3

Channe14

Addmon& SubtracUon

4883

8181

10963

32478

Mulllphcauon

3978

6668

89!0

26386

68

78

226

DtVLSIOR

40

Number of anthmettc operauons mvolved when applymg algonlhm S over channels 1-4

TableS 54

Root number

Channel!

Root 1

-131 00

-102 61

-168 89

-104 99

Root2

-108 10

-135 43

-168 80

-105 43

Root3

-106 98

-9792

-17176

-130 16

Root4

-----

-98 53

-103 22

-131 73

RootS

··--

Root?

---

---------

-125 29

Root6

------

RootS

Channel2

Channel3

Channel4

-lOO 83 -!01 54 -100 29

Roots accuracy obtamed usmg algonlhm S over channels 1-4

Operation

ChannelS

Channel6

Channel?

ChannelS

Addtuon & Subtractton

43200

32574

70547

72911

Multiplicauon

35032

26436

57006

59234

2!0

318

512

DtvtSIOn

264

Number of anthmeuc operauons mvolved when applymg algonthm S over channels 5-8

Root number

ChannelS

Channe16

Channel?

ChannelS

Root 1

-11184

-16193

-178 17

-129 52

Root2

-12163

-11697

-123 92

-117 10

Root3

-116 21

-118 32

-179 02

-173 04

Root4

-113 30

-133 74

-114 10

-119 61

RootS

-110 09

-118 48

-167 52

-107 79

Root6

-108 88

-130 99

-!08 82

-122 25

Root?

-124 48

-135 09

-11077

-107 78

RootS

-110 lO

-!05 99

-115 55

-127 69

-108 68

-105 90

-11423

----------

-105 37

-115 68

Root9

--

RootlO

--

Root 11

---------

Root 12

Root 13

TableS 57

Roots accuracy obtamed usmg algonthm 5 over channels 5-8

158

-!05 89

-118 89

-10617

-102 91

---

-11035

-----------------------------------------------------------------------------

CHAPTER6

ADAPTIVE DECISION FEEDBACK EQUALIZERS

6.1 INTRODUCTION In the transmission of digltal data at the highest possible transmission rate over a bandlimited telephone channel, it is normally necessary to employ an equalizer at the receiver to correct the signal distortion introduced by the channel [3,27,32]. The equalizer is usually implemented in the form of lmear transversal filters and can contribute a considerable increase to the cost of the modem, especially when a microprocessor system is used in the implementation of the modem [33,84]. This chapter first discusses the adaptive estimation of the sampled impulse response of the linear baseband channel and then extends the discussion to include the adaptive adjustment of the decision feedback equalizer. Three different algorithms are presented and tested by computer simulation, for the adjustment of the tap gains of the decision feedback equalizer (Fig. 6.1). The first algorithm adJusts the tap gams of the equalizer such that, at the exact equalization of the channel, the signal-to-noise ratio at the detector input is maximum. The remaining algorithms attempt to minimize the mean-square error in the equalized signal by means of gradient and Kalman techniques.

6.2 FEED FORWARD TRANSVERSAL FILTER CHANNEL ESTIMATOR The first of the basic techniques for estimating the sampled impulse response of the channel uses an adaptive feedforward transversal filter [3,32-38] and is a development of the technique used by Magee and Proakis to estimate the complex vector, Y, (sampled Impulse response of the channel). It has (g+l) taps which IS equal to the number of components of the sampled impulse response of the channel, and the tap gains are adjusted in such a way as to minimize the mean-square error

159

-

between the received sampler, (Fig. 6.1) and its estimate. When perfectly adjusted, the tap gains are the components of the sampled impulse response of the equivalent, baseband discrete-time channel model. The adaptive feedforward transversal filter channel estimator is as shown in Fig. 6.2 and operates as follows [3,34-38]. Each square marked Tin Fig. 6.2 is a store that holds the corresponding detected data symbol {.f,_J for h = 0, 1, ... , g. Each time the stores are triggered, the stored values are shifted one place to the right At time t=iT, the estimator IS fed with the received sample r, and the detected data symbol .f,. The detected data symbols {.f.} are assumed to be correct, so that.f, =s, for each

i. Let f, _1 be the previous stored estimate of Y, then an estimate of r, at the output of the estimator can be formed by

f.

=

...

6.1

...

6.2

The error in f., which is

e,

=

r, -

f,

is scaled by a small positive quantity, 1:1, to give M,. The resulting signalis multiplied by each of the information symbols .f~-h at the appropriate tap (where "' denotes complex conjugate). The resulting products are then added to the corresponding components of the previous estimate, f, _1, to give the new stored estimate f., whose (h + I)" component is

... 6.3

for h = 0, 1, .... , g. The above equation is usually known as the stochastic gradient algorithm (which is derived from the steepest descent algorithm) [35,85] for adjusting the tap gains of the channel estimator. It minimizes the average mean-square error between f, and r,. The above algorithm itself is usually known as the least mean-square (LMS) algorithm.

160

The factor!!. in Eqn 6.4 is the step size or (the averaging parameter of the estimator) and it need not necessarily be a constant. It is usually a fairly small number (of order

!x 1o-3 ) and determines the amount of adjustment made to the tap gains. The smaller the value of!!., the smaller IS the effect of additive noise on Y, but the slower is the convergence of Y, towards Y [37]. In the form presented, it can be seen that the number of complex multiplications involved in the generation of Y, is (2g+3) per received sample. The estimation process descnbed above is probably the least complex of all possible methods, and involves relatively few operations per iteration [3,17,34-38]. Clearly, the feedforward transversal filter channel estimator can be implemented easily, and it is able to track slow variations in the channels response [86]. However, it is well known that when the input samples {s.} are highly correlated, the convergence of the estimator to the optimum value is slow [87]. In the steady-state operation, the rate of change of the tap gains will be small. As such, after the initial convergence of the estimator has been achieved, the processor which performs the estimation process may be allowed to be idle, the actual adjustment of the tap gains only being performed after the error signal exceeds some pre-determined limit or after a given number of symbols have been processed and detected [34-38]. In order to evaluate the performance of the estrmator, computer simulation tests have been carried out over models of eight telephone channels, where the objective is to determine the convergence rate of the channel estimator. The sampled impulse responses of the equivalent linear baseband channels are as shown in Tables 2.1 and 2 2. The tap gains of the channel esttm!!tOr are adjusted dunng a training period of 500 symbols at the start of transmission, during which there is no need to detect the data symbols. At the start of the training period the tap gains are all set to zero, this condition representing no prior knowledge of the channel. At the end of the training period the tap gains of the channel estimator are frozen and then held fixed over the following data signal. The constant!!. (Eqn. 6.3) is given the value of 0 004 for the first 240 symbols of the training signal, the value of 0 001 for the next 100 symbols and the value ofO 00053 for the remaining 160 symbols in tests over channels 1-4. However, tests over channels 5-8, have shown that a better performance for the channel estimator can be achieved when the constant!!. is given the value of 0.001 forthe first 160 symbols, the value of0.0005 forthe next 150 symbols and the value

161

----------

of 0 00015 for the remaining 190 symbols. These arrangements approach the best overall adjustment of the channel estimator for channels 1-8, over a traimng period of 500 symbols, with~ being permitted to take on three different values. The measure of performance of the channel estimator was the squared error between the known sampled impulse responses of the time invariant channel, Y, and the estimated sampled impulse response, Y., which ts defined here as,

=

... 6.4

Fig. 6.3 shows the convergence rate of the channel estimator when estimating the sampled impulse response of telephone channel 1 at signal-to-noise ratios of 15, 20, 25, 30, 35, 40, 45, 50, 55 and 60 dB. Tests with channels 2-8, have shown that the convergence rate of the channel estimator is approximately similar to that achieved over channel!, especially when the number of components in the channel estimator is the same. Therefore, the results of the convergence rate of the channel estimator over channel 1 have been considered to be representative of all the channels under test. Furthermore, the value of '\l1 (l=i) at the end of the iterative process has been calculated and recorded as shown in Tables 6.1 and 6.2, respectively for channels 1-4 and 5-8. The signal-to-ratio is denoted SNR, where

... 6.5

bearing in rrund that the mean-square values of s,, 2 (?,are 10 and 42, respectively for channels 1-4 and 5-8 and the mean-square of w, is 2 Jmtlally, decreases and then remains fixed, especially when there is an insufficient number of taps in the filter D (30 taps or less).

iii-

At low signal-to-noise ratios (30 dB or less), the number of taps m the filter D has no significant effect on the value of\jf. However, at high signal-to-noise ratios, and with channels that introduce severe distortion, the effect of

165

--------------------------------------------------------------------------

increasrng the number of taps m the filter D becomes quite Significant. This suggests that at high signal-to-noise ratios and with channels that introduce severe distortion, the decrease in the value of 'V is due both to the channel estimator and lmearpre-detection filter adjustment method (more taps in the filter D are required). It has been found from the tests that if there is a slight change in the estimate of the sampled impulse response of channel 4, the root-finding algorithm might fat! to locate all the zeros off (z) that lie outside the unit circle. This, of course, leads to a significant increase in the value of 'Jf. The problem has been identified and solved by using nine starting points instead of five starting points in the root-findingalgorithm. The latter starting points are used with channels 1-4 whereas the former starting points are those suggested for channels 5-8 (Section 5.3.5). Throughout the tests, the number of arithmetic operations mvolved in the adjustment of the linear pre-detection filter, that is, addition, subtracuon, multiplication and dtvis!On of real numbers were computed and recorded as shown in Tables 6.4-6.18. The number of arithmetic operations involved in the estimation of the sampled impulse response of the linear baseband channel has not been included in the latter Tables, since this number is equal to (2g+3) complex multiplications per iteration. As further tests, the tolerance of the equalizer to additive white Gaussian noise over the eight models of telephone channels were obtained as shown in Figs. 6.4-6.9. It is assumed here that the equalizer has 50 taps and that there are g taps in the linear filter D and F, respectively. By transmitting around 1.5 x 106 data symbols per curve in Figs. 6.4-6.9, and m some cases muc~ more, it has been possible to keep the 95% confidence limits for the different curves to less than about ±0.5 dB [88]. Extra tests have were also carried out, but now an infrnite number of taps in the filter D and perfect estimation of the sampled impulse response of the channel were assumed. Furthermore, the roots of the z-transform of the sampled impulse response of the channel have been calculated theoreucally [60] The equalizer, adjusted as just described, is referred to as the equalizer la. In the form suggested here, equalizer la is a nonlinear equalizer with tap gains set to those presented in Tables 3.1 and 3.2 (ZF equalizer) for channels 1-8. In tests with equalizer la, the tolerance of the equalizer to additive white Gaussian noise was obtained and plotted (Ftgs. 6.4-6.9).

166

Extra tests, whose results are not considered here, have also been earned out to determine the effect of the number of taps in the channel estimator (value of g) on the performance of equalizer 1. In tests over channels 1-3, a significant improvement in the values of 'l>I> 'JI and a fraction of a dB increases in the tolerance to additive white Gaussian noise, together with a remarkable saving in the number of arithmetic operations mvolved, can be achieved by reducing the value of g to 15. However, the improvement has not been obtained in tests with other channels (channels 4-8) in spite of the significant improvement in the value of u 1• The degradation in the performance of the latter channels, arising from reducing the value of g, is due to the fact that the z-transform of the sampled impulse response of those channels have more than eight roots lying outside the unit circle in the z-plane. These roots require more taps in the pre-detection filter to equalize, especially when their magnitudes close to unity. Furthermore, reducing the value of g will obviously degrade the roots location accuracy as seen from 5.3.1. Finally, in the tests with equalizers 1 and la over channels 7 and 8, it has been found that both equalizers suffer from very long error bursts. However, the results presented in Tables 6 3-6.18 show that the technique employed in equalizer 1 correctly adjusts the tap gains of the filters D and F. This suggests that the decision feedback equalizer can no longer be recommended as a detection process. It also suggests that a better tolerance for the system to additive while Gaussian noise can be achieved when the adjustment technique is combined with a more sophisticated detector, such as near maximum-hkelihood detector [17 ,31,51,55,59]. 6.3.2 Equalizer 2 This is based on the least mean-square stochastic gradient algorithm, which is extensively reported in the literature and is widely applied m practical communication equipment [3,25,27 ,32-33,89]. The adjustment ofthe tap gains of the linear feedforward transversal filters employed by the decision feedback equahzer is such that the mean-square error in the equahzed signal is minimum. This can be considered as the conventional practical approach towards the minimum mean-square error equahzer. This equalizer uses the error in the equalized signal 6.13

167

----------------------------------------------------------------------------------

to adjust the tap gains {dh} and {fi.} of the filters D and Fin Fig. 6.1. It is assumed here that the lmear filter Din the forward section is a linear feedforward transversal filter With (n+ 1)-taps and the linear filter Fin the feedback section is also a linear feedforward transversal filter with g taps. In the adaptive adjustment mode, and after the reception of r., the coefficients of the filters are adjusted recursively, in order to follow time variations m the channel response. This is done as follows. The error signal given in Eqn. 6.13 is first calculated. This error signal is then scaled by a factor (step size) 11,, and the resulting product is then used to adJUSt the coefficients of both fllters according to the steepest-decent algorithm [3,25,27 ,32] ;

for

h=O,l, . .

. ,n

...

6.14

and

J..,

for

h=l,2, . . . ,g

... 6.15

where dh.• and f. .• are the (h + 1)'h and the hrh tap gains, respectively, of the filter D and F and where* denotes compl,ex conjugate. The parameter 11, is a small posiuve constant which rmght be fixed or variable. The rate of convergence towards the desrred adjustment of the equalizer increases with 11,, but the accuracy of the adjustment decreases as 11, increases. Thus a suitable compromise between the two conflicting requirements must be accepted. The problem of interest here is whether or not it is, in fact, possible to achieve the reqmred setting that minimizes the mean-square error in the equalized signal by means of the gradient algorithm within a reasonable time. Computer simulation tests have been carried out to determine the performance of the equalizer when operating over the eight models of telephone channels. In all tests, the tolerance of the equalizer to additive white Gaussian noise was obtained. The tap gains of the equalizer are adjusted by means of a known random training signal of 10,000 and 13,400 symbols, respectively, for channels 1-4 and 5-8, at the start of the transmission, dunng which there is no need to detect the data symbols.

168

--

At the start of the training signal the tap gains are all set to zero, this condition representing no prior knowledge of the channel. At the end of the training signal the tap gams are frozen and held fixed over the duration of the data signal. The same signal-to-noise ratio is, of course maintained over the training signal as over the data signal. In tests over channels 1-4 the constant !'J., (Eqns. 6.14 and 6.15) is assigned the value 0.002 for the fust 3000 symbols of the training signal and the value 0.0002 for the remaining 7000 symbols. However, in tests over channels 5 and 6 the value of the constant is set to 0.0004 for the frrst 3000 symbols and 0.00005 for the remaining 10400 symbols. This arrangement approaches the best overall adjustment of the equalizer for channels 1-4 and 5-6, with a training signal of 10,000 and 13,400 symbols, respectively and with !'J., being permitted to take two different values. Tests with channel 7 have shown that it is possible to obtain a satisfactory adjustment of the equalizer only if the length of the training signal exceeds 25,000 symbols, but this number is far from acceptable in any practical system design. Furthermore, m tests over channel 8, it has not been found possible to achieve a satisfactory adjustment of the equalizer even when !'J., takes different values and the traimng signal exceeds 50,000 symbols. Ideally, under the assumptions made here, equalizer 2 can be adjusted as closely as required to minimize the mean-square error in the equalized signal, so long as the training signal is made sufficient long. In practice, however, telephone circuits are not strictly time invariant and, moreover, they introduce various transient effects, such as impulsive noise and occasional signal-level and carrier phase changes [5,27,90], which temporarily degrade the adjustment of the equalizer. The latter is also affected, to some degree, by errors in the detected data symbols [91]. Furthermore, even quite small drifts in the sampling phase of the received signal require a significant adjustment in the equalizer tap gains, thus setting an upper limit on the adjustment of the equalizer [92]. Since tests in which equalizer2 was adjusted during the detection of the data have led to inconclusive results, it was finally decided to adopt the procedure descnbed, in which about four seconds under reasonably ideal conditions are allowed for the adJUStment of the equalizer. The tolerances of the equalizer to white Gaussian noise have been obtained and plotted as shown in Figs. 6.4-6.9, respectively for channels 1-6. The number of taps

169

in filters D and F are the same as those used by equalizer 1. The results of equalizer 2 over channels 7 and 8, have not been presented here, smce it is not possible to achieve a satisfactory adJustment of the equahzerwithin four seconds trainingpenod.

6.3.3 Equalizer 3 A limitation of equalizer 2 is that it does not make full use of all the mformation available to it at the time of adaptation, making its rate of convergence relanvely slow [3,17,32-33]. More sophisticated techniques, such as theKalman filter can be used to overcome these limitations [3,17,32-33,93-94]. The Kalman algorithm is designed to update the tap gains of the linear feedforward filters employed by the equalizer, adaptively, from the received and previously detected signals at its input. The tap gains are adJusted such that the mean-square error in the equalized signal is a minimum [3,32,33,93-94].

r,_ 1

Letr,

r,_.and.f,

•••

§,_ 1

•••

.f,_ 8 betheinputtothefeedforwardand

the feedback sections of the equalizer, respectively, at t=iT. The input signals can be represented by the vector X,

= [

r,

r,_ 1







r,_.

s,

.f,_ 1







s,_ 8

]

•••

6.16

Also let C,_ 1 be an estimate of the tap gams of the equalizer at time t=(i-1)T

c,_l

= [

d,-1.o

d,-1.1

.

.

.

d,_l.•

f-1.1 f.-1.2 . . . f.-1.. l

... 617

The signal at the output of the equalizer at timet =(I-1)T is C, _1X,r, (where T denote the transpose), which may be different from the ideal output by an error e., where

...

6.18

It can be shown that the new estimate of the tap gams of the equalizer that mmim1zes the mean-square error in the equalized signal is

170

g~ven

by [3,32,93-94]

C,

=

C,_t

+

G,e,

...

6.19



...

6.20

...

6 21

where

• G,

P,_tx.r

=

ro+X,P,_tx;

and

P,

=

1 (J)

(P,_I

The symbol G, is a

-

G,X,P,_J

(n+g+l) · -component vector known as the Kalman gain and P, _1 is

the (n+g+ 1) x (n+g+ 1) covariance matrix of the error in the estimate of the actual equalizer coefficients. The parameter ro in Eqn. 6.21 and 6.22 is a positive constant less than unity, usually very close to unity, selected to provide short term averaging. Initially, the matrix P, is set to the identity matrix and C, is set to a null vector. Further details on the algonthm and on the derivation of Eqns. 6.19 and 6 21 are given elsewhere [3,17,32-33,93-94] As before, the performance of equalizer 3 was tested by computer simulation and the results of these tests appear m Figs. 6.4-6 9, respectively, for channels 1-6. In all tests, the tolerance of the equalizer to additive white Gaussian noise was obtained and plotted. It is assumed that the filter D and F has 50 and g taps, respectively. The value of the constant ro in Eqns. 6.20 and 6.21 was set to 0.9999. Furthermore, the tap gains are obtained by transmitting a known and random sequence of 10,000 symbols. After this training period, the tap gains are frozen and held fixed. Tests with channels 7 and 8 have shown that, although a correct adjusnnent of the equalizer setting has been achieved with a mean-square error ofless than -20 dB, the equalizer suffers from long error bursts. Therefore, the results showmg the tolerance of equalizer 3 over channels 7 and 8 have not been considered.

171

6.4 ASSESSMENT OF RESULTS AND DISCUSSION The results presented in Figs. 6.4-6.9 show the tolerance of the equalizers to additive white Gaussian noise. The relative merits of the equalizers can now compared in the light of these results, as follows; i-

The performance of equalizer 1 compared with that of equalizer la shows that there IS a degradation (a fraction of a dB) in the equalizer performance over channels 1-3. However this degradation increases slightly over channels 4-6. The degradation in the equalizer 1 over channels 1-3 is mamly due to the channel estimation error, u 1 whereas over channels 4-6 it is due to both channel estimation error and the fimte number of taps in the filter D. Therefore, a better performance over channels 1-3 can be achieved using a more sophisticated channel estimator whereas with channels 4-6 a better performance can only be achteved by using a better estimator together with a larger number of taps in the filter D. In the latter case, a larger number of taps are needed because channels 4-6 have more roots lying outside the unit circle than channels 1-3, and that some of these roots are very close to the unit circle, as shown in Tables 4.1-4.2.

ii-

At an error rate of 10-4, the performance of equalizer 2 is always inferior to that of equalizer 1. This degradation increases as the level of the distortion increases. However, at an error rate of 10-1, the performance of equalizer 1 becomes inferior, especially over channels introducing severe distortion (channels 4-6). Over typical channels, both equalizers achieve approximately the same performance.

m-

The performance of equalizer 1 over the range of the error rates tested in Figs. 6.4-6.9 is as good as that of equalizer 3 for channels 1-2 (the typical channels), but is inferior for channels 3-6 (severely distorted channels). However, the performance of equalizer 1 improves as the error rate decreases. Furthermore, for channels 1-6, equalizer 1 appears hkely to be at least as good as, or better than equalizer 3 at error rates of 10-6. The latter conclusion is derived from an extrapolation of the results presented in Ftgs. 6.4-6 9, since tests cannot be carried out at such low error rates.

172

iv-

The performance of equalizer 2 is always inferior to that of equalizer 3. The degradation in the performance of equalizer 2 over the error rates tested, is generally less than 1 dB for channels 1-3, but increases up to 5 dB at lower error rates for channels introducing more severe distortion (channels 4-6).

The tendency for the performance of equalizer 1 to improve, relative to that of equalizers 2 and 3, as the error rate in the{§,} decreases, can be explained as follows. At high error rates (> 10-2), the predominant factor that determines the error rate, for a given signal-to-noise ratio in the equalized signal, is the mean-square distortion in the latter signal. Thus, equalizers 2 and 3, which attempt to reduce this distortion, can be expected to operate well under these conditions. The latter effect

IS

very

significant with channels 4-8, where the mean-square distortion is very high. However, at low errorrates (< 10_.), the predominant factor that determines the error for a given signal-to-noise ratio in the equalized signal, is the peak distortion in the latter signal. In fact, the large maJority of all errors now occur with the sequences of data-symbol values that lead to the most severe distortion in one or more of the corresponding equalized signals. Although the mean-square error caused by equalizers 2 and 3 is small, the peak distortion can be quite large. Equalizer 1, on the other hand, (assuming the correct detection of the preceding data symbols which is normal under the assumed conditions), introduces no distortion. Thus the increased noise level in the equalized signal of equalizer 1 relative to the others is now more than offset by the relatively large distortion that is experienced from time to time in the equalized signal of equalizers 2 and 3. In the computer simulation tests over channels 1-3, it was found that the tap gains of equalizer 2 approximate quite closely to the corresponding taps of equalizer 3. This

IS

consistent with the good performance of equalizer 2 over those channels,

and suggests that for channels that do not introduce extremely severe distortion, the gradient algorithm is a cost effective technique for adjusting the equalizer because of its computational simplicity. However, over channels introducing severe distortion (such as channels 4-6) the performance of equalizer 2 is inferior to that of equalizer 3, between 2 to 5 dB. Furthermore, the gradient technique did not achieve a satisfactory adjustment of the equalizer tap gains over channels introducing more

173

---------- - - - - -

severe distortion, such as channels 7 and 8. Therefore, over channels introducing severe distortion, the gradient algorithm is no longer a useful technique for the adjustment of the equalizer. Equalizer 1 achieves a performance as good as that of achieved by equalizers 2 and 3 over typically distorted channels 1-3. However, over channels 4 and 6 the performance of equalizer 1, compared to that of equalizer 3, IS inferior by about 1 dB. This degradation in the performance mcreases as the distortion increases to about 2 dB in channelS. But equalizer 3 is considerably more complex than equalizers 1 and 2. However, vanous techniques are now available for simplifying the practical implementation of equalizer 3 and a substantial reducnon in complexity can be achieved by such methods [3,32,95-96]. Unfortunately, these techniques have a tendency towards a steady build-up in round-off errors, which can cause the system to become unstable [ 17]. Furthermore, even with such simphfications, the technique is still complex compared to that employed in equalizer 1. Therefore, it is clear from the above discussion that over channels introducing a typical amount of distortion, the technique employed in equalizer 1 can be considered for further development, whereas with channels introducing very severe distortion the decision feedback equalizer is no longer a suitable detection process. Fmally, a detailed study is now under investigation to develop the technique employed in equalizer 1 using the TMS320 family of DSP (Digital Signal Processor). Details on this study can be found elsewhere [97].

174

Linear baseband channel Y L,s,o(t-iT) Receiver filter

Transmission ath

{s,) Detector

~ to ..

Lioear filter D

Detected data symbols

{q,}

'

~

Lioear filter F

\..e r(t) ~-Sampler Received t = 1T

-

Linear filter adjustment method

.......

F

_t_y ~"'

Clhannelestimator

.

Fig. 6.1 Model of data transmission system

signal

A

T

1----__,._.

T

s,_2 ----------------·-~

A

Yl-1,0

Y1-1,1

C. e.

! - = . : c _ - - - f - - - - 1 - - - - - f - - - -.....--------------·---_.,

Fig. 6.2 Linear feedforward transversal filter channel estimator

Y1-1,g

0

-10

m

-20

"Cl

£

.,-

177

L.

-30

........ 0

c: 0

~

-40

0

E ;;

LJ

-50

-60

-70+-------.------.-------.------,-------.------.------~-------.------~----~ 0

50

100

150

200

250

300

350

Number of symbols

Fig. 6.3 Channel estimator performance over channel 1

400

450

500

~\

0,1

~~ \~

~\ ~~\ ~~\

0.01

-. CD

E

.,.... 0

~~\

0

.0

E

>-

\\~\

(I)

0,001

~\.

Legend

• • D

0

Equalizer 1 Equalizer~

Equalize.r~

0.0001 10

\\\.

;qualiZE!.!:.,2_

12

\\

14

16

18

20

Signal-to-noise ratio in dB

Fig. 6.4 Performance of equalizers with channel 1

178

22

0.1

'\~~

~

0 01

~\, ~~,'\ ~~\ ~\,\ ~\\

-".... .;

.... 0

.,........ 0

.0

E

>-

IJ'l

0.001

Legend

• •

Equalizer 1

D Equalizer 1a

0

0.0001 10

~ualiz~2Equalizer~

12

~

14

16

18

20

22

Signal-to-noise ratio in dB.

Fig. 6.5 Performance of equalizers with channel 2

179

,~, '~, \\

0.1

~

'\

0.01

\

-"' ....0

.... 0 ....

\

\\

\\\\, \'',\\

"'0

.n

E

>-

VJ

0.001

\\,\

Legend

• •

Equalizer 1

~

D Equalizer _!Q

0

~ualiz~2Equalizer~

\ \ \ ~

\ \ 0.0001 18

20

24

22

26

Signol-fo-noise rofio in dB

Fig. 6.6 Performance of equalizers with channel 3

180

28

0.1

~\

\

~

\\ \\ \

\

\

\ \ \

\ \ \

\ \ \ \\

0.01

-.

\

~

... e "' 0

.0

E (f)

I

\

\

I

0.001

Legend •

\

D Equalizer 1a

e

~ualiz~2-

0

Equalizer~

\ \

\

\



I

Equalizer 1

\

\ I I

\

\ \

\

I

\I\I

\ \

\ \

0.0001+------,,-----,----t\lE,-I"I---r----.\--.... ---, 36 u 26 28 ~30 32 Signal-to-noise ratio in dB

Fig. 6.7 Performance of equalizers with channel 4

181

\ ~,

0.1

\

\

\

\

\

\

\

\

\ \ \ \ \ \ \

\ \ \\\ \ \\ \ \\ \ I\ \

0 01

-"......

I

(I)

...0 0"

\

I

..a

E

>-

(/)

\ \ \

I

0.001

Legend



Egualizer 1

~ualiz~2-

0 Egualizer l_

0.0001 26

28

\ \

I

0 Egualizer 1a



\

30

\ \

\

\ \

I

\ \ \

\

\

I

32

\ 34

36

38

Signal-to-noise ratio in dB

Fig. 6.8 Performance of equalizers with channel 5

182

40

-------·-----------------------------------------------------------------

0.1

0.01

-... .,;

~

...e " 0

\

.0

E

\

;;;-

\

\

0.001

\

Legend •

\

Equalizer 1

\

D Equalizer .!Q

e

~ualiz~2-

0

Egualize.r ~

\ \

\ \

26

28

30

32

34

36

38

Signal-to-noise ratio in dB.

Fig. 6.9 Performance of equalizers with channel 6

183

-

s:-...'R

Channel!

Channel2

Charmel3

Channel4

IS

-270

-270

-25 8

-253

20

-329

·32.9

-32.0

-319

25

-353

-35 4

-33 8

-33 7

30

-396

-397

-38 7

-38 I

35

-445

-446

-432

-431

40

-495

-497

-47 8

-46.7

45

-541

-54 2

-52.8

-53 4

so ss

-58 3

-574

-54 0

-581

-642

-637

-57 3

-53 6

60

-608

-62.3

-57 8

-561

Table 6 I

Error in !he estimate of !he sampl ed impulse response of channels 1-4.

51\'R

ChannelS

Channel6

Charmel7

ChannelS

IS

-242

-24 7

-21 3

-249

20

-29 6

-302

-266

-319

25

-344

-34 7

-31 2

-35 I

30

-37 s

-38 4

-35 8

-386

35

-42.1

-42.4

-40 I

-42.6

40

-477

-479

-39 4

-483

45

-52.8

-53 4

-456

-53 6

so ss

-503

-52.0

-406

-585

-58 0

-60 8

-42.7

-62 I

60

-593

-612

-449

-665

Table62

Error in !he esumate of !he sampl ed Impulse response of channels 5-8

-

184

Channel2

Channell

Channel3

Channel4

SNR

ljl,

ljl,

ljl

ljl,

ljl,

ljl

ljl,

ljl,

15

-32.1

-266

-25 s

-30 8

-27 s

-25 8

-28 3

-22.5

-21

20

-405

-319

-314

-38 6

-32.7

-317

-35 I

-28 4

ljl,

ljl,

ljl

s

-19 9

-17 3

-15 4

-27 s

-2M

-24 6

-22.4

ljl

25

-42.9

-344

-33 8

-42.4

-34 7

-341

-38 0

-30 I

-294

-300

-252

-23 9

30

-440

-394

-381

-42.1

-409

-38 s

-39 I

-35 9

-342

-29 6

-300

-26.8

35

-491

-442

-430

-508

-442

-344

-452

-397

-38 6

-32.2

-339

-300

40

-53 7

-491

-47 8

-537

-498

-483

-487

-446

-432

-31 8

-38 4

-309

45

-59 7

-52.9

-52.1

-58 4

-53 7

-52.5

-54 s

-492

-481

-29 8

-45 I

-29 6

so ss

~81

-551

-SS 4

~8

-551

-SS I

~16

-493

-491

-299

-477

-29 8

~7.7

-59 s

-58 9

~s

~0

-59 I

-519

-53 4

-52.1

-325

-442

-32.3

60

-71.8

-573

-571

~88

-58 8

-58 4

~10

-532

-52.5

-303

-466

-302

Table 6 3

Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and lmear ft.lter D m equahzer 1 usmg 30-t.aps m the lmear ft.lter D

Operat10n

Channel!

Channel2

Channel3

Channel4

Addtuon & Subtracnon

6908

10997

14027

38891 31504 366

Muluphcauon

5662

8968

11418

DtVISlOR

152

184

202

Table 6 4

Number of anthmettc opeanons mvolved m equallZer 1 wtth 30-taps m the lmear fllter D.

ChannelS

Channe16

ChannelS

Olannel7

SNR

ljl,

ljl,

ljl

ljl,

ljl,

ljl

ljl,

ljl,

ljl

ljl,

ljl,

ljl

15

-23 I

-216

-192

-23 0

-23 I

-20 I

-17 0

-11 8

-10 6

-10 9

-12.7

-87

20

-292

-263

-245

-260

-282

-240

-15 3

-16 s

-12.8

-170

-13 I

-14 s

25

-289

-306

-267

-209

-32.0

-205

-19 4

-19 8

-16 6

-15 7

-18 s -13 8

30

-25 I

-32.5

-244

-197

-353

-19 6

-22.2

-24 0

-200

-210

-20 8

35

-31 8

-382

-309

-19 9

-397

-19 9

-22.2

-29 I

-21 4

-8 8

-243

40

-24 8

-42.1

-24 7

-19 9

-449

-19 9

-22.7

-270

-21 3

-16 s -28 0

-16 2

45

-212

-47 s

-212

-19 3

-Sl 0

-19 3

-14 9

-287

-14 7

-10 5

-33 2

-105

-17 9 -8.7

50

-23 9

-451

-23 8

-19 2

-48 4

-19 2

-19 6

-283

-19 I

-13 3

-357

-13 3

55

-205

-52.6

-205

-19 3

-58 0

-19 3

-246

-292

-23 9

-13 4

-416

-13 4

60

-21 I

-53 7

-211

-19 3

-584

-19 3

-17 6

-285

-17 6

-46

-478

-46

• Table 6 S

Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and hnear fJ.lter D m equahzer 1 usmg 30-taps m the lmear ftlter D

Operatton

ChannelS

Channel6

Channel?

ChannelS

Addtuon & Subtractton

45683

40773

63353

76822

Multtpltcatton

37095

33097

51225

62425

Dwtslon

384

362

414

622

Table 6 6

Number of arithmetic opeanons mvolved m equalizer 1 wtth 30-taps m the lmear fllter D

185

Channel!

Chatmel2

Channel3

Channel4

SNR

v.

ljl,

'V

v.

v,

'V

v.

'V>

'V

v.

v,

'V

IS

·31 3

-25.7

-24 7

-300

-26 8

-25 I

-27 6

·22.5

·213

-20 I

·17 8

·IS 8

20

-42.5

-31 8

-314

·38 7

·32.7

-31 7

-343

·283

-273

-234

·23 2

·203

25

-41

s

·352

·342

-413

-35 6

·346

·372

-304

·296

·29 4

·25 4

-240

30

-443

·39 I

·38 0

-42.5

-405

·38 3

·391

-35 2

-337

·248

·295

-23

35

-485

-438

-42.6

·516

-437

-430

-460

·392

-38 4

·377

·33 7

·32.2

40

·551

-497

-487

.ss 8

-500

-490

-514

-445

-437

·28 4

-38 9

-280

45

-606

·53 4

·52.6

·59 6

-540

·53 0

·559

-492

-484

·307

-439

-305

50

-67 2

-562

·55 8

·64 6

·562

-556

-62.8

·500

-498

·302

-459

-30 I

55

-677

-59 I

·58 s

-681

·59 6

-59 0

·58 7

·561

·542

·291

-459

·290

60

-72.9

-58 I

-580

-68 8

·591

-587

-647

·53 3

-53 0

·293

-487

·29 3

Table 6 7

s

Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and lmear ftlter D m equahzer 1 usmg 40·taps m the lmear Illter D

Operat10n

Channel!

Channel2

Channel3

Addlt!On & SubtractiOn

7258

11001

14487

39234

Multtpltcauon

5942

8956

11778

31732

Dtvtston

172

200

222

382

Table 6 8

Channe14

Number d. anthmeuc opeanons mvolved m equalJ.Zer 1 wtth 40-taps m the hnear ftlter D

ChannelS

ChaMel7

Channel6

ChannelS

SNR

v.

ljl,

'V

v.

v,

'V

v.

'V>

'V

v.

ljl,

'V

IS

·24 9

·21 7

·200

-25 6

-226

-209

·18 I

·11 7

.lQ 8

·13 0

-8 7

·13

20

·29 5

-25 6

-242

-28 8

-274

·250

-204

·14 7

·13 6

·11 3

-17 s ·14 4

25

-33 3

·292

·27 8

·32.9

-30 5

-285

·22.5

-19 6

·17 8

-16 6

-17 3

·13 9

30

·366

-32.6

·312

-30 I

·352

·289

-23 6

·24 I

·20 8

-20 I

·19 7

·16.9

35

-39 3

-38 9

·36 I

·34 6

-406

·33 6

·23 7

-300

-22.8

·225

-24 6

-204

40

·360

-42.3

-351

-352

455

·349

-18 9

·28 7

·18 5

-206

·28 5

·19 9

45

·32.7

-476

·326

-369

·5Q5

·367

·209

-315

·205

·15 0

-34 6

·14 9

50

·31 0

-455

-309

-373

-48 3

-37 0

-171

-269

·16.7

-13 4

·377

·13 3

55

·362

·519

·361

-35 4

-56.1

·35 4

·31 4

-277

·26.1

·111

-41 8

-17 I

60

-307

-506

·306

-352

-58 3

·35 2

·166

-313

·16 8

·214

-48 9

-214

Table 6 9

Dtscrepancy between the actual and the esumated sampled tmpulse response of the channel and hnear ftlter D m equahzer 1 usmg 40-taps tn the lmear f'tlter D

Operatton

ChannelS

Channel6

Channel?

ChannelS

Addlllon & Subtractton

41493

37016

82478

73821

Multlphcatton

33631

29969

66576

59873

DtVlSlOil

372

512

610

Table 6 10

. 350

Number of anthmettc opeanons mvolved m equaliZer 1 Wllh 40-taps m the lmear fllter D.

186

Channel2

Chatu1ell SNR

IV·

IV>

Chatulel3

IV

IV·

IV,

IV

IV.

IV>

Channel4

IV

IV·

IV

IV·

15

-31 3

-246

-23 8

-302

-25 4

-242

-220

-266

-207

-203

-17 6

-15 8

20

-428

-313

-31 0

-387

-322

-31 3

-276

-34 2

-26.8

-27 0

-227

-213

25

-428

-35 6

-34 8

-428

-36.0

-351

-306

-372

-29 8

-299

-25 5

-242

30

-456

-39 9

-389

-437

-41 0

-39 2

-357

-407

-34 5

-32 3

-299

-279

35

-493

-454

-439

-513

-453

-443

-403

-459

-392

-403

-340

-33 I

40

-55 6

-485

-477

-54 5

-490

-47 9

-436

-53 9

-432

-35 8

-38 4

-33 9

45

-615

-52 9

-523

-595

-53 6

-526

-49 I

-56.4

-48 3

-411

-436

-39 2

50

-667

-566

-562

-64 8

-56 8

-562

-517

-655

-515

-406

-47 I

-397

55

-677

-584

-57 9

-715

-597

-594

-51 4

-607

-509

-412

-442

-39 4

60

-68 9

-58 5

-58 I

-671

-587

-58 I

-495

-593

-490

-41 0

-429

-38 8

Table 6 11

Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and lmear filter D m equahzer 1 usmg 50-taps m the lmear ftlter D

Operat10n

ChaMe11

Channel2

Channe13

Addtuon & Subtracuon

7608

11461

14946

42083

Multiphcatton

6222

9316

12138

33994

Dtvtston

192

220

242

416

Table 6 12

Channel4

Number of anthmeuc opeanons mvolved m equalLZer 1 wtth 50-taps m the lmearfllter D.

ChannelS

ChaMe17

Channel6

ChaMelS

SNR

IV,

IV· -21 6

-10 6

-10 I

IV· -13 0

IV

-202

IV· -19 6

IV,

-19 5

IV· -257

IV

-243

IV· -212

IV>

15

-10 8

-87

20

-307

-25 5

-243

-307

-27 5

-25 8

-229

-169

-15 9

-17 9

-17 6

-14 7

25

-336

-28 7

-27 5

-30 I

-30 I

-271

-23 0

-19 3

-17 8

-19 7

-16 6

-14 8

30

-365

-33 5

-31 8

-343

-36.1

-321

-254

-24 0

-217

-212

-221

-18 6

35

-31 9

-390

-312

-386

-407

-365

-221

-313

-216

-227

-259

-21 0

40

-318

-425

-314

-371

-46.1

-366

-19 3

-304

-18 9

-25 4

-28 8

-23 8

45

-35 6

-474

-353

-47 5

-4~9

-455

-19 4

-341

-19 3

-18 0

-35 2

-17 9

50

-33 5

-503

-33 4

-527

-528

-498

-24 3

-287

-23 0

-15 7

-357

-15 7

55

-36.7

-53 3

-366

-511

-54 4

-494

-19 5

-299

-19 I

-245

-41 I

-244

60

-367

-524

-366

-512

-569

-502

-18 7

-33 3

-18 5

-26 8

-51 I

-26 8

Table 6 13

IV

IV

Dtscrepancy between the actual and the estunated sampled tmpulse response of the channel and lmear filter D m equaltzer 1 usmg 50-taps m the lmear filter D.

Operatton

ChannelS

Channel6

Channel?

ChannelS

Addttton & Subtractton

46530

46073

83485

85284

Multtphcatton

37678

37290

67354

69125

DtVISIOQ

418

424

Table 6 14

534 -

7fJ2

Number of anthmettc opeanons mvolved m equaltzer 1 Wtth 50-taps m the lmear filter D

187

Channel!

Channel2

Channe13

Channel4

SNR

'I'I

'i'l

Ill

'I'I

'i'l

'I'

'Ill

'Ill

Ill

'Ill

'Ill

Ill

15

-319

-251

-24 3

-30 8

-26.6

-247

-271

-22.3

-21 0

-19 9

-17 6

-15 6

20

-402

-31 0

-305

-37 8

-317

-30 8

-352

-26.7

-26.1

-260

-22.4

-208

25

-449

-346

-342

-453

-34 8

-34 4

-37 8

-304

-297

-309

-25 4

-24 3

30

-443

-396

-38 3

-437

-403

-38 6

-409

-357

-34 6

-337

-306

-28 9

35

-507

-440

-432

-52.6

-441

-436

-46.3

-38 9

-38 2

-404

-32.8

-321

40

-563

-485

-478

-551

-492

-482

-53 4

-439

-434

-389

-38 I

-35 5

45

-603

-541

-53 2

-60 I

-544

-53 4

-55 6

-495

-486

-449

-430

-409

50

-650

-569

-563

-647

-571

-564

-669

-529

-52.7

-448

-46.2

-424

55

-689

-582

-57 9

-714

-59 8

-59 5

-603

-543

-53 3

-417

-453

-401

-691

-591

-587

-690

-604

-59 8

-53 8

-506

-489

-42.9

-432

-400

60

Table 6 IS

Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and hnear ftlter D m equabzer 1 usmg 60-taps m the lmear fllter D

Operatton

Channel I

Channel2

Cbannel3

Channe14

AddtUon & Subtractton

7958

12377

15674

47433

Mulupllcauon

6502

10048

12716

38290

Dtvtston

212

244

264

468

Table 6 16

Number of anthmeuc opeanons mvolved Ill equahzer 1 wuh 60-taps tn the hnear filter D

Cbannel S

Channel6

Channel?

ChannelS

SNR

'Ill

'i'l

Ill

'Ill

'Ill

Ill

'Ill

'Ill

Ill

'Ill

'Ill

Ill

15

-24 5

-204

-19 0

-25 5

-209

-19 6

-18 7

-Ill

-104

-12.9

-10 I

-8 3

20

-29 8

-263

-24 7

-302

-286

-263

-23 6

-16 5

-15 7

-16 8

-16 6

-13 7

25

-341

-28 8

-277

-33 6

-295

-28 0

-245

-19 6

-18 4

-19 9

-17 5

-15 5

30

-352

-34 5

-31 8

-37 I

-36.8

-340

-29 6

-257

-242

-215

-206

-18 0

35

-33 7

-403

-32 8

-426

-419

-392

-25 6

-29 I

-23 9

-19 3

-251

-18 3

40

-34 5

-42.1

-33 8

-432

-460

-413

-281

-24 7

-23 I

-22.3

-28 9

-215

45

-42.2

-476

-41 I

-50 8

-504

-476

-259

-33 4

-252

-22.4

-35 8

-22.2

50

-38 2

-484

-37 8

-55 5

-542

-51 8

-32.6

-30 I

-28 I

-15 3

-361

-15 3

55

-421

-50 9

-41 5

-55 8

-55 3

-52.5

-26.1

-28 6

-242

-16 8

-379

-16 8

60

-39 4

-54 2

-393

-602

-579

-559

-283

-33 2

-271

-303

-51 0

-303

Table 6 17

Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and bnear filter D m equaliZer 1 usmg 60-taps m the lmear ftlter D

Ope rat ton

OtannelS

Channel6

Cbannel7

ChannelS

Addttlon & Subtractton

46157

38435

72709

86179

Multtpltcauon

37322

31003

58603

69751

Dtvtston

430

386

502

718

Table 6 lS

Number of anthmettc opeanons mvolved m equaliZer 1 With 60-taps tn the lmear ftlter D

188

CHAPTER 7 ADAPTIVE ADJUSTMENT OF THE DIGITAL RECEIVER WHEN OPERATING OVER HF RADIO LINKS 7.1 INTRODUCTION The radio frequency band in the region of 3·30 MHz is traditionally known as the High Frequency (HF) band and has for many years been used as a transmitting medium for long distance communication systems [3,5,7,17-24]. HF transmission is unpredictable due to the existence of multipath propagation and Rayleigh-fadmg [3,7]. However, the use of an HF channel as a transmission medmm is still of great importance, inspite of the introduction of transmission media such as satellite and optical fibre links, since HF links are both economical and flex1ble. In a high speed serial data transmission system operating over a 3 kHz band in the HF band spectrum, one of the major problems facing the modem designer is that of eliminating the intersymbol interference caused, by multipath propagation [3,7,98-99]. Therefore, when it is required to transmit data at the h1ghest poss1ble rate, it is necessary to employ an adaptive equalizer fortrackmg the sampled impulse response of the resulting time-varying baseband channel [3,17,22,98-99]. Traditionally, the tap gains of the equalizer have been adjusted adaptively using the gradient, Kalman or latice algonthms, to minimize the mean-square error in the equalized signal [3,17,32-33,98-99] . .Alternatively, an equalizer may be adjusted from the estimate ofthe sampled impulse response of the channel [17,27,39]. This chapter is pnmarily concerned with the latter type of equalizer. 7.2 MODEL OF SYSTEM Fig. 7.1 shows the model of a synchronous serial data transmission system which employs an HF radio link as the transmission path. The information to be transmitted is coded onto a set of data symbols {s,}

189

s, = s... + jsb,

."

7.1

...

7.2

where

S

a,,,

Sb,J

=

±1 -

or

±3

--J=T. The {s.}, is stansucally independent and equally likely to have any of their possible values. It is assumed that s, =0 for ig. The filter Fin the feedback section of the data receiver is a linear feedforward transversal filter whose tap gams at time instant t=(i+n)T are given by the g-component vector Ym+h,h = 0

for

193

. f. •• ..]

7.14

F, +• can be considered as an estimate of the sampled Impulse response of the channel

and filter D, which should be provided at the detector mput by the adaptive adjustment algorithm Ideally, when the filter D is correctly adjusted,

f. ••. h

e,.. n+h for

h = 1, 2, ... , g. With the correct detection of the data symbols and at the time instant t=(i+n)T, the output Signal from the filter F is an =

accurate estimate of the mtersymbol interference present in v, • ., such that the equalized signal is x, +•· Under these conditions

7.15

where q, •• is the output signal from the filter Fat t=(i+n)T. The detector itself then operates on the equalized signal which is: 7.16

to determine the detected data symbol

s,.

Three different fading channels have been studied and these are referred to as the HF channels 1-3. The signal over channel 1 is transmitted via three sky waves whereas the signal over channels 2 and 3 is transmitted via two sky waves. The frequency spread for channels 1 and 3 is 2 Hz, whereas for channel2 It is 1 Hz. The transrmssion delays

't 1 and 't2

of the sky waves relative to that of the shortest delay

sky wave, together with the other channels parameters, are as shown m Tables 7.1, where 0.4167 is the period Tin rmlh-seconds. Figs. 7.4-7 .6, show the characteristics of channels 1-3 over a duration of 25 seconds of transmissiOn.

7.3 ADJUSTMENT ALGORITHMS This section is concerned with the adaptive adjustment of the digital data receiver (filters D and F) shown in Fig. 7 .1. In Its ideal form the pre-detection filter is adjusted to be an all pass network, such that the resultant sampled impulse response of the

194

channel and filter is minimum phase [27 ,39]. Funhermore, the filter is such that in the resultant z-transform, all zeros of the z-transform of the sampled impulse response of the channel that lie outside the umt circle in the z-plane are replaced by the complex conjugate of their reciprocals, the remaining zeros bemg left unchanged [27 ,39]. Section 4.4 together with Section 5.3.5 descnbe a possible arrangement for the adjustment for the filters D and F shown in Fig. 7.1 when operating over time-invariant channels. However, when operating overtime-varying channels, such as those considered here, it is no longer possible, even with an infmlte number of taps, to achieve the ideal adjustment of the filter. In the tests here, it is assumed that the linear filter D has 50 taps whereas the filter F has g taps (g=29). It is also assumed that the channel estimator correctly estimates the sampled impulse response of the linear baseband channel such that;

Y,

=

Y,

7.17

Further details on the method for estnnatlon of the sampled impulse response of the HF link are given elsewhere [17 ,20,34,38,104]. Finally, tests have shown that the location and equalization of roots with absolute values greater than 1.05, rather than unity, degrades the performance of nonlinear equalizer by only a fraction of a dB [105]. However, it can be shown, that with this process, a remarkable saving in the number of arithmetic operations can be achieved, since the location of roots with absolute values less than 1.05 requires more iterations and a large number of taps are also required to equalize such roots. Therefore, 1t will be assumed throughout the following stydy that the adjustment algorithms equalize only those roots with absolute values greater than 1.05. Computer simulation tests have been carried out with the HF channels 1 and 2 using different algorithms for the adjustment of the filters D and F to achieve the objective described earlier. Three different sequences have been selected to represent the fadmg introduced by the given channels, to test the adjustment algonthms. These sequences are as follows, i-

Sequence 1 applies to channel 1. It has 400 values of Y, ranging from i=500 to i=900 (Fig. 7.4). Over the given sequence the value of I Y.l (measured in

195

dB) decreases to -7 dB and then increases again. The number of roots with absolute values greater than 1.05 increases from 5 to 7 and decreases to 5 again as shown in Fig. 7. 7. The exact number of roots has been calculated theoretically using the appropriate NAG library routine [60].

ii-

Sequence 2 is applies tochannel2. It has 400values off, ranging from i=3300 to i=3700 (Fig. 7.5). Over the sequence, the value of IY.l decreases to -12 dB and increases again. Six roots with absolute values greater than 1.05 appear simultaneously to g~ve a total of nine roots With absolute values greater than 1.05 during the sequence, as shown in Fig. 7.8.

iii-

Sequence 3 also is applies to channel2. It has 400 values of Y, ranging from i=5200 to i=5600 (Fig. 7 .5). During the sequence the number of roots with absolute values greater than 1.05 decreases rapidly from 9 to 3 as shown in Fig. 7.9.

7.3.1 Adjustment algorithm 1 The algorithm is that proposed in Section 6.3.1 with the same mne starting points as suggested for the telephone channels 5-8, to locate the roots of Y, (z) with absolute values greater than 1.05. It has been found that the algorithm with the given set of starting points always fails to locate all the required roots. Furthermore, the missing roots are usually very close to the unit circle but in certain cases the absolute values of the missing roots exceeds 1.25. The missing roots can have a serious effect on the performance of the adaptive receiver [105]. ThiS suggests that the nine starting points which have been employed for the telephone channels must be modified for the HF channel. In testing algorithm 5 (Section 5.3.5) over telephone and HF channels, it has been found that most of the roots of Y,(z) have being located using the first starting point (0.0

+ jO.O). Additional tests over HF channels have shown

that a better performance of the root-finding algorithm (that is, fewer missed roots), can be achieved when the last eight of the starting points move closer to the unit circle. The new set of nine starting points is as shown in Table 7.2. These points have been proposed by D. H. Lehmer [72], as centres of eight circles used to cover the unit circle, in checkmg whether or not there is a root inside the circle. However, tests have shown that a better performance can also be achieved by moving the eight

196

"

I ' !

starting points more closer to the unit circle than that suggested earlier in Table 7 .2. The latter set of starting points is referred to as set 2 whereas the former set is referred to as set 1. Tests over the HF channels 1 and 2 have shown that set 2 gives a slightly better performance than set 1, as can be seen from Figs. 7.7-7.9 and Tables 7.3-7.5. Tables 7.3-7.5, show the average number of arithmetic operations involved in the algorithm by the use of set 1 and set 2. 7.3.2 Adjustment algorithm 2 It has been found that, when operating over a time-varying channel, each of the roots at the time instant t=iT are very close to the corresponding roots at the time instant t=(i-1)T. Therefore, a considerable saving in the number of arithmetic operations involved in the adjustment algorithm can be achieved by using the roots at the time instant t=(i-1)T to locate the roots at time instant t=iT. This comes from the fact, supported by computer simulation tests, that these roots require very few iterations (1 to 3) for convergence to be achieved. However, there are roots appearing and disappearing from the region outside the unit circle. This suggests that the use of the previously located roots alone, for the given starting points from which the roots are located will not make any improvement to the performance of the algonthm. However, a novel arrangement have been proposed by S. F. Hau, which uses the previously located roots, has enabled the algorithm to give much improvement in its performance together with large saving in the number of arithmetic operations involved [106]. This arrangement can be described as follows; i-

In the first run of the algonthm (i=1), the value of~ is set to the frrst of the starting points in set 2 (Table 7 .2) and the algorithm starts looking for the roots as described in Section 5.3 .5. When the whole set of mne starting points has been used and the algonthm has not converged to a root, the algorithm will be terrrnnated and it will be assumed that all k roots are located. The values of the negative of the reciprocals of the located roots,{~m(i)}, will be stored in an array together with the nine starting pomts of set 2, as shown in Fig. 7.10.

u-

In the second run of the algorithm, (i=i+ 1), the value of~ is set now to the first value m the starting set given in Fig. 7.10, ~ 1 (1). This will converge very

197

quickly to its corresponding value,

p1(i + 1). If the roots corresponding to

P1(i + 1) are greater than 1 05 the value of P1(i + 1) will replace P1(i) in Fig. 7.10 otherwise the values of P1(i) will be discarded from the set given in Fig. 7.1 0. The next step is that the algorithm sets A. to ~(i) and this, of course, will converge to Its corresponding value of Pii + 1) and the process will be repeatedk times. After k such repetitions the algorithm will use the remaining nine starting points as described earlier. This of course first checks whether any more roots are left outside the required region and at the same time attempts to locate the given roots. If this check results in a root with modulus greater than 1.05, the negative of its reciprocal wiii be stored together with - the previous k{PmCi + 1)} in the starting array. The procedure just described is clanfied by the flow diagram given in Fig. 7 .11. Computer simulation test have been carried out with the sequences 1-3, using the algorithm 2 to adjust the tap gains of the filters D and F. The results of these tests show the number of roots tracked by the algorithm together with the average number of arithmetic operations involved in the algorithm and are given in Figs. 7.7-7.9 and Tables 7.3-7.5. The results clearly demonstrate the great saving in the number of arithmetic operations. The algorithm successfully tracks all the roots of sequence 3, whereas with sequences 1 and 2, it fails to locate all the required roots. However, the root-trackmg capability is much better than that of algorithm 1.

7.3.3 Adjustment algorithm 3 In the tests ofalgorithm2 with the sequences 1-3, it has been found that occasionally the algorithm fails to locate some of the roots of Y,(z) whose absolute values are greater than 1.05. Tests have shown that, at certam instants with the HF channels 1-3, the absolute values of the missing roots are as high as 1.16. As mentioned earlier, these missing roots can seriously degrade the performance if their absolute values become any larger. To improve the root-tracking capability of algorithm 2, a modified version to algorithm 2 is proposed. In the new arrangement the algorithm attempts to locate all the roots of Y, (z) that he outside the unit circle, using the same procedure as that descnbed in the previous section, but the roots with absolute values

198

greater than 1.05 are the only roots to be equalized. In other words, the latter roots are the only roots to be replaced by the complex conjugate of their reciprocals and to be used for the adjustment of the tap gains of the linear filter D. Algorithm 3 was tested in the same manner as algorithms 1 and 2 with the sequences 1-3, and the results of these tests appear in Figs. 7.7-7.9 and Tables 7.3-7.5. The results clearly show that algorithm 3 successfully locates most of the roots of Y,(z) with absolute values greater than 1.05, while algorithms 1 and 2 fail to locate some of these roots. The results presented in Figs. 7.7 and 7.9 show that algorithm 3 successfully locates most of the roots with absolute values greater than 1.05, when operating with the sequence 2. The results presented in Fig. 7.8 also show that, although the algorithm fails to locate all the required roots, the root-tracking capability is much better than that achieved by algorithms 1 and 2. However, the results presented in Tables 7.3-7.5 show a significant increase in the number of arithmetic operations involved in the algorithm compared with that required by algorithm 2. 7.3.4 Adjustment algorithm 4

Algorithm 4 is an improved version of algorithms 2 and 3. The approach considered here is to reduce the average number of arithmetic operations involved in algorithms 2 and 3. As mentioned earlier, algorithms 2 and 3 use the last nine starting points in the array shown in Fig. 7.10 to check whether any roots with absolute values greater than 1.05 appear or not and at the same time to locate these roots, if any. By observing the change in the number of r_oots With absolute values greater than 1.05, it has been found that most often the number is fixed for a very long period and sometimes this period exceeds 1000 samplmg instants especially over 2 sky waves HF channels (channels 2 and 3). Furthermore, it has been found that during such a period, the number of operations involved in using the remaining nine starting points in the array is the same as that involved when using the algonthm with the fust k startmgpoints in the array given in Table 7.10 to locate the roots. It should be borne in mind that the use of the remaining nine starting pomts here only gives a check that there are no more roots with absolute values greater than 1.05.

199

A useful reduction in the average number of arithmetic operations can be achieved through the use of the Schur algorithm described in Section 5.2.1, to check whether there is a root with a modulus greater than 1.05 [73]. This root-identification algorithm has been combmed With algorithms 2 and 3 as follows; i-

Initially, the new algorithm starts looking for the roots in the same way as before. The values of the k {~m(i)} are also stored in an array as shown in Fig. 7.10.

ii-

In the second run of the algorithm and after the whole set of k{~m(i)} will

be used to look for their corresponding values of k{~m(i + 1)}, the Schur algonthm is applied to F 1 (z ). If the Schur algorithm returns a decision that

F 1 (z) has no roots with absolute values greater than 1.05, the algorithm terminates without using the remaining set of nine starting points. However, if the Schur algorithm returns a decision that F 1 (z) has a root with a modulus greater than 1.05, the remaining set of nine starting points is used as for the algorithms 2 and 3. Computer simulation tests have also been carried out with the sequences 1-3 and algorithm 4. The results of these tests, which show the average number of arithmetic operations involved with the sequences 1-3, are given in Table 7.3-7.5. The results here assume that the Schur algorithm is combmed with algorithm 2. It has been found that when the Schur algorithm is combined with the algorithm 2, the same number of roots are tracked as by the algorithm 4. The same result also holds when the Schur algorithm is combined with the algonthm 3. Again, when using the Schur algorithm, the number of components in F 1 (z) is truncated to 10. This drop in the number of components, as shown in Section 5.2.1, has no effect on the decision reached by the Schur algonthm. It has also been found that algorithm 4 requires slightly more arithmetic operations than algorithm 2, especially when the Schur algonthm locates a root. However a great saving m the number of operations is achieved when the Schur algorithm decides that there are no more roots with absolute values greater than 1.05. This can be clearly observed when comparing the results presented in Tables 7.3-7.5. The selected sequences are of course those where roots cross the circle ofradius 1.05.

200

· 7.3.5 Adjustment algorithm 5 In testing of algorithm 2 with the sequences 1-3, it has been found that a useful reduction in the average number of arithmetic operations involved can be obtained by applying the adjustment algorithm every 4, 8, 12 and 16 sampling instants, as shown in Tables 7.6-7.8. The same saving, in the average number of arithmetic operations will, of course, also be achieved by the use of algorithms 3 and 4. The tests here also involve measurements of the error introduced in the sampled impulse response of the channel and the pre-detection filter by the given process. The measurement used are 'Jf2 and 'Jf3, where 'Jfz is as defined in Eqn. 6.11 and

7.18

The parameter 'Jf3 is a measure of the error introduced in

v,..

caused by the

components e, .. ,0, e,+., 1, • • • , e,+•.•-t• which should be all zero. The error here is normalized by taking it relative to the resultant of the magnitudes of

e..... , e, .. ,-+ 1, •• , e,+•.•+• [17]. The evaluation of 'Jfz and 'Jf3 for every sampling instant is complex and at the same time does not reveal much information about the performance of the algorithm, especially in a long test Therefore, the tests have considered only the worst values for 'Jf2 and 'Jf3 given by the algorithm. The results presented in Tables 7.9-7.11 demonstrate the effect on the values of 'Jfz and 'Jf3 , when algorithm 2 IS applied every 4, _8, 12 and 16 sampling instants, with the sequences 1-3. Similar effects are also observed when algorithm 3 or 4 is applied every 4, 8, 12 and 16 sampling instants. Although the sequences 2 and 3 have SIX roots crossing the circle of radius 1.05 simultaneously and sequence I has only two roots appearing and dtsappearmg, t}:le increase in the values of 'Jfz and 'Jf3 is much greater than with sequence 1. This is because sequence 1is given by a 2Hz frequency spread HF channel, whereas sequences 2 and 3 are given by a 1 Hz frequency spread HF channel From F1gs. 7.4-7.6, it can be seen that, with channels 1 and 3 (2Hz frequency spread) I Y,l changes more rapidly than with channel 2. It is clear from the results presented in Tables 7.9-7.11, that the effect of increasing the period, over which the algorithm is idle, has a greater influence on 'Jfz than those on 'Jf3•

201

A further useful reduction in the average number of anthmetic operations together with a significant decrease in the value of 'lf2 and 'lf3 compared with what presented earlier can be achieved through the application of prediction to the roots of Y,(z) that are being tracked by algonthms 2,3 or 4. Algorithm 5, based on such proposal, is a combination of degree-1 least square fading-memory pred1ct1on and algorithm 2 [17,20,38,104,107-108]. The algonthm locates and predicts the roots as follows; i-

Initially, let

Pm Cl)

and

P~(l) be the negative of the rec1procals of the mm

evaluated orpred1cted roots ofY,(z). Furthermore, let p~(O) = 0 form= 1, 2, ... , k.

il-

At the instant t = iT, apply algorithm 2 as descnbed m Section 7.3.2 to

detenmne the values of kfPm(l)} iu-

After the time mstantjT the degree-1least square fadmg-memory prediction is applied todetenmne the successive values of k{PmCi)}, k{P~(I + j, i)}. The latter can be ach1eved as follows; (1)

The error signal between the evaluated Pm(!) and its predicted value at time mstant t=(l-2J)T is calculated as ...

(2)

7.19

Following the calculation ofthe prediction error the one-step pred1ction of the rate of change of the PmCi) with respect to i is calculated using [17 ,20,38,104, 107-108]

...

7.20

where c1 is a posit1ve real constant in the range from 0 to 1, usually very close to 1 [17,20,38,104,105-106].

202

---- -------

(3)

Having calculated the rate of change of ~.. (i), the predicted value of~.. for the time instant t=(i+2j)T can be updated using ..

(4)

7.21

For the adjustment purposes it is necessary to know the values of k{~..} at time mstant t=(i+j)T. This can be achieved using

~~(i+j,i) = ~~(i+2j,i) iv-

-

~~~(i+2j,i)

...

7.22

At t=(i+j)T, the values of k{~~(i + j, i)} are used to adjust the tap gams of the linearfilterD and the values ofk{~~(i + j, 1)} form=!, 2, ... , k are stored in array as shown m Fig. 7.10.

v-

After time instant jT, algorithm 2, 3 or 4 is applied with the starting points returned from the previous step to determine the k values of {~.. (i + 2j)} and steps ii-iii are repeated.

Fig. 7.12 shows the logic flow diagram of algorithm 5 when the roots are evaluated and predicted every 4 sampling instants. Algorithm 5, has been tested by computer simulation and the results of these tests showing the worst values for"ljf2

and . "ljf3 over sequences 1-3 are shown in Tables

7.12-7.14. Throughout the tests the average number of arithmetic operations were calculated and these are recorded m Tables 7.15-7.17. It is assumed through the tests that degree-1least square fading-memory prediction is combined with the algorithm 2. Furthermore, the roots are evaluated every 16 samplmg instants, and are predicted every 16 sampling instants, but interleaved halfway between the measurements, to give an adjustment every 8 sampling instants. The constant c1 has the value of 0 92. The same improvement is gained when algorithm 3 or 4 is combined with the prediction algorithm.

203

It is clear from the results presented in Tables 7.12-7.17, that algonthm 5 saves a significant number of arithmetic operations and at the same time keeps the value of

'Jf2 as good as that achieved when algorithm 2 is applied. Furthermore, the effect on the value of 'Jf3 is as that when algorithm 2 is applied in those of prediction to give an adjustment every 8 sampling instants. The most difficult task in adJUStment using algorithm 5 is in the optimization of the value of c1, since the optimum value changes from channel to channel and from sequence to sequence.

7.4 DECISION FEEDBACK EQUALIZERS OPERATING OVER HF CHANNELS Here, the study presents different types of results which consider the effect of additive noise on the algorithms performance. The decision feedback equalizer discussed in Sections 3.4.1 and 6.3.1 was used here to investigate the performance of the algorithm. Several algorithms and techniques for the adjustment of the tap gains of the decision feedback equalizer were employed in the test. It is assumed that the estimate of the sampled impulse response of the channel and linear filter D has been employed by the hnear filter F. It is also assumed that the feedback filter has 29 taps. In order to obtain the most accurate comparison possible between the algorithms, the same fading sequence given by the values of the sampled impulse response of the linear baseband channel in Fig. 7.1 was used over the given number of 40,000 received samples {r,} that were employed in each test, but not, of course the same noise sequence. The equalizers under test were obtained by employing the ideal algorithm, algorithm 2 and algorithm 3 in the adjustment of the decision feedback equalizer. These equalizers are referred to here as HI, H2 and H3. In the equalizer HI, it IS assumed that the number of taps m the linear filter D is infmite whereas the number of taps in the feedback filteris g. It is also assumed that the all the roots of Y, (z) lying outside the unit circle are equalized. The latter roots were found theoretically using the appropriate NAG library routine [60]. In the equalizers H2 and H3, it is assumed

204

that algorithms 2 and 3 are used in the adjustment of the equalizers. It is also assumed in the equalizers H2 and H3 that the linear filter D has 50 taps and the roots to be equalized are only those with absolute values greater than 1.05. The results of computer simulation tests showing the relative tolerances of the equalizers to additive white Gaussian noise when operating over HF channels I and 2 are as shown in Figs. 7.13 and 7.14,respecuvely. The signal-to-noise ratio (SNR) in dB is given by

...7.23

bearing in mind that the mean-square value of s,

and

w, are 10 and 2cr'-,

respectively. Finally, the effect of applying the algorithm 2 every 8 or 16 sampling instants on the tolerance of the equalizer H2 to additive white Gaussian noise was also investigated and is as shown in Figs. 7.15 and 7 .16, respectively forthe HF channels I and 2. In Figs 7.15 and 7.16 the equalizers H2A and H2B, refer to equalizers where algorithm 2 is applied every 8 and 16 sampling instants, respectively. Furthermore, equalizer H5 refers to the equalizer obtained by using algorithm 5 in the adjustment of the tap gains of the equalizer. It is assumed in the latter equalizer that algorithm 5 evaluates or predicts the roots every 8 sampling instants. The most Important conclusions obtained from Figs. 7.13-7.16 can be summarized as follows; i-

The degradation in the performance of equalizer H2 and H3 over that ofH1, increases as the signal-to-noise increases. This degradation in the performance of the equalizers H2 and H3 is caused by the following factors, (1)

Equalizers H2 and H3 equalize only the roots ofY, (z) with absolute values greater than 1.05, whereas H1 equalizes all the roots ofY, (z) that lie outside the unit circle.

205

(2)

The limited number of taps in the linear filter D, which can causes some problems in equalizing the roots of Y,(z) that are oflower absolute values and when the number of roots of Y,(z) with absolute values greater than 1.05 is high.

(3)

Algorithm 2 or 3 in the equalizers H2 or H3, respectively occasionally failed to find all the roots off,(z) with absolute values greater than I 05.

(4)

The problem of selecting the best delay in Y., from Eqn. 7.12, for evaluating the tap gains of the fllters D and F. The above factors mdicate the reasons for the greater degradation in performance of equalizers H2 and H3 over channel 2, when compared with their performance over channel!. This degradation occurs because, with channel2, there are six roots with absolute values greater than 1.05 appearing and disappearing as shown on Figs. 7.8 and 7.9, during a relatively short period. To be equalized, these roots usually require more taps in fllter D, and at the same time, it is possible that the root-finding algorithm fails to locate some of these roots.

ii-

At high error rates, the dominant factor that produces errors in the detected data symbols is additive noise. Therefore, it is difficult to distinguish which algorithm is better at these error rates. However, at low error rates, the dominant factor is intersymbol interference. This can be greatly reduced by the use of better adjustment algorithms and more taps in the filter D.

iii-

From the results presented in Figs. 7.15 and 7.16, it is clear that, at very high error rates, it is possible to apply algorithm2, every 8 or 16 sampling instants without any fear of any undue performance degradation. However, at low error rates it is possible to apply algorithm 2 every 8 or 16 sampling instants, only with channel2, whereas with channel!, every 8 sampling mstants seems to be the !unit This suggests that the smaller the frequency spread, the longer is the period over which the adjustment algorithm can remain idle. Of course, with channel 2, the variatton of Y,(z) with i is much slower than that of channel!, as shown in Figs. 7.4-7.6.

206

1v-

The equalizer HS achieves, as expected, a performance which is better than that of equal1zer H2B and worse than that of equalizer H2A, as shown in Figs. 7.15-7 .16. At high error rates the performance of the equalizer HS is as good as that achieved by equalizers the H2 and H2A. However, at low error rates, the performance of the equalizer HS deteriorates when compared with that of the H2 and H2A equalizers. The deterioration is much more in channel 1 than in channel 2, due to the larger frequency spread.

7.5 ASSESSMENT OF THE RESULTS AND DISCUSSION i-

In tests of algorithm 2 with the sequences 1-3, it has been found that a considerable improvement in the value of \j/2 can be achieved through the use ofY•-•+A (Eqn. 7.4) in place ofY, (Eqn. 7.6), m the adjustment of the tap gains of the linear filters D and F at llme instant t=iT. The results of these tests also show that the improvement 1s dependent on the value of h. It has been found that the best value for h depends on the sequence (or the channel) under test, and it usually is in the range 3 to 6, as shown in Tables 7.18-7.20, respectively for sequences 1-3. It has also been found that the average number of arithmetic operations involved is the same as that when using the true Y, (Eqn. 7.6).

ii-

In tests of the equalizers H2 and H3 over channel 3, it has been found that incorrect operation of the adaptive filter occurs periodically. In parucular, there are now errors in the detected data symbols, even in the complete absence of noise, with a typical error rate of up to about 10-3 in the detected

-

data symbols. The errors occur during the deeper fades of the shortest duration, that is, wllh the most rapid fading rate. Different HF channels with the parameters sirmlar to that of channel3 also give the same results. During the deep fade in channel3, the following points have been observed; (1)

A sharp increase in the values of\j/2

and \j/3 , over a deep fade with a

peak error at the peak of the fade. However, at this instant, errors occur in the detected data symbols, even in the complete absence of the n01se.

207

(2)

During the period of the deep fade, the values of the four Gaussian noise wave forms(p 1(t),

pz(t),

p3(t)

and

p4 (t) Fig. 7.3) cross the zero

axis as shown in Fig. 7.17. (3)

The value of IY,l drops to -33 dB, which is such that, over a period of 50 samplmg intervals, the received signal level decreases by 12 dB to its lowest level and then increases agin by 12 dB.

(4)

As shown in Fig. 7.8, six roots with absolute values greater than 1.05 appear simultaneously during this deep fade, to give a total of nine roots with absolute values greater than 1.05. Tests with Y, --+h in place of that Y, with h havmg different values have shown that the error free cond!tion cannot now be achieved.

iii-

A further test with a different arrangement of algorithm 2 has been carried out. In this test, the value of the mean-square error in the equalized Signal was calculated and recorded as shown in Tables 7.21 and 7.22 for channels 1 and 2, respectively. The value of the mean-square error in the equalized signal is given by

000

7.24

The results clearly show the effect of applying algorithm 2 every 8 and 16 sampling instants on the value of

E.

It has been shown that there is an

improvement when algorithm 5 is evaluating or predicting the roots every 8 sampling instants compared with that of when algorithm 2 is evaluating the roots every 16 samplmg instants. Furthermore, the tests also show the effect of using Y,_n+h in place of Y,. It has been found that when his equal to 5, a lower mean-square error can be obtained, so that h=5 is assumed in Tables 7.21 and 7.22.

208

Linear baseband channel

,.............................................................................................................................................................................................................................................................................................................................. . .

:

:

I,s,o(t-iT)

:

' ~-,-.,..-+~

Transmitted data symbols

! !

Transmitter filter

Linear modulator

HF radio

link

Linear demodulator

Receiver filter

: :1

: 1-4-__.,

''' White Gaussian noise '' '' .............................................................................................................................................................................................................................................................................'' A

( s,) Detector

~

.

De tected data symbols

rB¥ (q,l

~

Linear filter

F

.... ~

Fi

-

( r,}

...." -

Linear filter D

tDi

Linear filter adJustment method

tvi Channel estimator

Fig. 7.1 Model of data transmission system

'

sampler t =tT

r(t) Recetved signal

•vu

80

~

m

~

60

" 0

., .,"

:;: ::J

-

40

0

213

-5

-"" ""0

·;:

m 0 ::::;: -10

-15

-20+-----------r-----------r---------~r---------~r----------.-----------,

0

10000

20000

30000

40000

Sampling Instant !IT!

Fig. 7.5 2 Sky waves Rayleigh fading HF channel 2

50000

60000

10

m ""0

.!: -10

-" >0

214

-c ""0 :I

-20

Ol

:::." -30

-40+-----------.-----------.----------,,----------,-----------,-----------, 60000 10000 20000 30000 50000 40000 0

Sampling Instant !IT!

Fig. 7.6 2 Sky waves Rayleigh fading HF channel 3

10

B

,,

0"

e .....0 ...

215

..r-

.a "

~

z

·-rj

6.

11 11

.--J

I I I

Legend

4

•::--------L

,,':!

•,

,,., ''•,

11

I

I

,;

1l

•:

I 11 :

:

I

I ' III : : ~ la 11 ~ I II : : 11 : ' Jl- - - - - ____ ,.__11- - - -11 ~ I I I ;~---.-.-1I

: :

:

.• J_,.. ____ _!::

I

t

..........!

I , ~I I I , 11 1 I: 11,111 I

~~.!!! !~

• ,

: :

u •:

--

• ............................. • -------------NAG roots

0 Set 1 roots

Set 2 roots

2

0 Algor[thm 2_roots t:. Algorithm 3 roots

0 0

100

200

Sampling Instant {lTI

Fig. 7.7 Variation of the number of located roots over sequence 1 for different algorithms

300

' 400

10

i j' i ·-,·-------;.----.----· : !r--·--·:r-·r:r:__ 1_J__________ . . ___ ,_________ ,

8

-I 11

'

" 15

e

216

r____!r'•f:':_ ___________f.!. __ .!.':... 1\

6

I

....0 .,...

_j

~--'-->

E

z"

.

4

L. •,.r----..l ............ .

J

,.......... ,......

:.:

II I

:

. .......

::..

::

11

1

,................,;

:

11~

:

l

_JI

.

I

.0

I I

::

-

.----------, L egend



...•

NAG roots

1 roots 0 Set.......................

e Set 2 roots ---------------



0

Algor9thm

~

roots

!:::. Algorothm 3 roots

0~--~----~--~~~--~ 300 400 0

100

200

Sampling Instant

!JT!

Fig. 7.8 Variation of the number of located roots over sequence 2 for different algorithms

10

I. r"

I I

.-·-···· .......... •••• J ••• -.

B

.

1;

....e

6

:

,,

I'

I I

"-----,

..\

p---- - ················-·

I I I 1-1 I I I

............ ·t .....

"---:-,I I I

l-----------,

0

217

... "

..a

54

Legend

z



.

..................

NAG roots

I I I IL,_ I I I I

-

......

D Set-1 roots ...........

• -------------0 Set 2 roots

2

Algor\thm 2. roots

t:. Algorithm 3 roots 0 0

100

200

Sampling Instant !IT!

Fig. 7.9 Variation of the number of located roots over sequence 3 for different algorithms

300

400

r

The negative of the rcciprocals' r The fixed nine startmg points '

(

y

of the prevtously located roots

Value

~I

Order

I

(set 2) given m Table 7.2

I

k

2

k+l

2

9

k+2

k+9

Fig. 7.10 Starting points arrangement in algorithm 2

i= 0

J

•• I= 1+

1

ReadY1

No

Yes

i= I

; Use the nine startmg

Use the starung points

pomts set 2 gtven m

shown in 710 to locate the k ~m}

Table 7 2 to locate

F•r

~t' ~2'

•••••

~.

.

+ Dtscard the value 5f ~m or wluch I~ I< m I 05 and store the remaming

Store ~ , ~ , ••. , ~. in array togl;ther wttli the nme startmg pomts as shown m Ftg 7.10

ogether With the rune start.J.n

pomts as shown in Ftg. 7 10

+

+ Use the values ofk(~m} to adJUS! the tap gams of

hnear fi1tcrs D and F

+ Fig. 7.11 Flow diagram of algorithm 2 218

\

----------------------------------------------------------------------------

i=O,J=4 IC=3

+ i =I+ I

+ Ready;

+ IC=IC+ I NO

IC=S

NO

YES

IC=4

YES em(l) = J3m(1)- J3~(1,i-2j)

Apply algonthms 2, 3 or 4

+ -



2

J3m(•+2J,1) = J3m(l,i-2j) +(l-c1) em(~

+ J3~(1+2J,l) = J3~(1,1-2J) + J3~(i+2j,l) + (1-c~em(•)

Store the evaluated roots (J3m(•)) m array as shown m Fig. 7.10

+

. . I . J3m(l+J,i) = J3m(i+2J,1)- 2 J3m(i+2j,l)

+

IC=O

J

Use tl1e evaluated roots or tl1e pred1cted roots to adJust the lap gams of the filter D

+ Fig. 7.12 Flow diagram of algorithm 5

219

~ \

0.1

\

~

\\ \\

\\ \\ '\ '\

0.01

-"....

\\

I..

.... 0

\\ '\ '\

0 .a

E

>-

Vl

\\

0.001

1\

Legend



Equalizer H1



~uallz~H~

\

I \ I \

0 Equalizer H2

\\

0.0001

18

20

22

\

24

28

28

30

32

34

SNR in dB

Fig. 7.13 Performance of equalizers over HF channel 1

220

~

0.1

~

~

~

"""'"~ -

"~

"~

\\ \\

0.01

\~

\\

"....

~

0

....

\\ \\ \ \

....0

0"

.0

E

>-

Vl

\\

0.001

\

\

Legend

• •

Eguallzer Hl ~uallz~H~

0.0001 20

22

2'

\ \ \

\\

D Egualizer H2

18

\

I\ I \ 26

28

30

32

3'

38

SNR in dB

Fig. 7.14 Performance of equalizers over HF channel 2

221

---

-------------------------------------------------------

0.1

'\

'\

'\

'\ \

0.01

\

\\~\

\\

~\

-."......

~\ \

\\ ~\ \' \I \ \\\I \ \\\ I \ \\\ I

.... 0

0

.a

\

E

>Ill

0.001

Legend

\



Equalizer H2



fuualizE!!:. H28

\

0 Equalizer H2A 0 Equalizer HS

0.0001 18

20

22

2•

26

28

30

32

3


224

o.oo

a."' 0

-0.05

c--

.... --- --~-

E

Vl

-----Legend •

-0.10

- --t

P,(t)

D M!) __

• E...(IL 0 p,(t) -0.15-+--------...---------,--------.-----------, 200 0 50 100 150

Sampllng Instant !IT!

Fig. 7.17 Variation of pJt) with i

Channel

Number of sky waves

Relauve transmtsston delays of sky waves 1'1 and 'tz (ms)

Frequency spread

00,1.1,3 0

2

~----+-----------+-----~ 3

~--~2---4------~2-------4~--------0 0,3 0 3

2

I

00,30

2

Channels used m the tests

Table71

No

Set 1

Set2

1

0 0000 + J 0 0000

0 0000 + J 0 0000

2

08999 +)00000

09091 +JOOOOO

3

00000 - J 0 8999

00000 -)09091

4

00000 +j08999

00000 +)09091

5

-0 8999 + j 0 000

-0 9091 + J 0 0000

6

0.5890 - J 0 5890

06428 -j06428

7

0.5890 + J 0 5890

06428 +j06428

8

-0 5890 + J 0 5890

-0 6428 + J 0 6428

9

-0 5890 - J 0 5890

-0 6428 - J 0 6428

Table72

Startmg pomt sets 1 and 2 used m a1gonthm 1

Operauons

Algonthm 1 (set 1)

Algonthm 1 (set 2)

Algonthm2

Algonthm 3

Algonthm 4

Addtuon & Subtraction

20914

22163

13564

14564

10622

Multtphcauon

17202

18220

10800

11964

8535

Dtvtston

264

272

209

217

187

Table? 3

Average number of anthmeuc operations mvo1ved over sequence 1.

Algonthm 1 (set 2)

Algonthm 2

Algonthm3

Algonthm 4

1)

AddtUon & Subtraction

12187

12941

14790

18731

14974

Muluphcauon

12941

10710

12124

15339

11943

DtVlSIOR

217

222

214

234

209

Algonthm 2

Algonthm 3

Algonthm4

Operauons

Table 7 4

Algonthm 1 (set

Average nt.nTlber of anthmeuc operattons mvolved over sequence 2

Algonthm 1 (set

Algonthm 1 (set

1)

2)

Addttton & Subtracuon

23464

31743

15798

21998

12570

Multtphcatton

19279

26029

12937

17997

9969

DtvtSIOR

279

326

217

256

189

Operattons

Table 7 5

Average number of anthmettc operattons uwolved over sequence 3

225

- - - - - - - - - - - - - - - - - - - --

Operauon

EveryT

Every 4T

Every ST

Every 12T

Every 16f

Add!uon & Subtracuon

13147

3566

1818

1216

931

Mult!phcatton

10800

2927

1492

997

763

DtVISIOD

209

54

27

18

13

Table 7 6

Average nwnberof anthmeuc operations mvolved over sequence 1 when algomhm 2 applied every 1, 4, 8 or 16 samplmg mstants

Operatton

Every T

Every 4T

Every ST

Every 12T

Every 16T

Addmon& Subtracuon

14790

3615

1960

1353

1025

Mulupltcatton

12124

2966

1606

1109

840

DtvlSton

214

54

27

19

14

Table 7.7

Average number of anthmeuc operaUons mvolved over sequence 2 when algonthm 2 apphed every 1, 4, 8 or 16 samplmg mstants.

Operatton

EveryT

Every 4T

Every ST

Every 12T

Every 16T

Add!uon& Subtracuon

15798

3977

2155

1441

1110

Multlphcatton

12937

3256

1763

1179

908

DtviSton

217

54

28

18

14

Table7 8

Average number of anthmeuc operattons mvolved over sequence 3 when algonthm 2 apphed every 1, 4, 8 or 16 samplmg mstants

Parameter

Every T

Every4T

Every ST

Every 12T

Every 16T

"'' "''

-2850

-27 54

-2516

-2036

-17 04

-3250

-3123

-28 67

-26.38

-24 51

Table7 9

The worst values of 'Vz over sequence 1

and

'!f3 when afgonthm 2 apphed every 1, 4, 8 or 16 samplmg mstants

Parameter

Every T

Every 4T

Every ST

Every J2T

Every 16T

"''

-3140

-3138

-31 38

-31 38

-28 34

-31 82

-31 80

-31 60

-3160

-3125

"''

Table710

The worst values of 'lfz over sequence 2

and

'l'l when algonthm 2 applied every 1, 4, 8 or 16 samplmg mstanu

226

-

p arameter

Every T

Every 4T

Every 8T

Every 12T

Every 16f

'~'•

-3097

-30 11

-3011

-3015

·26.78

'If,

-3496

-2689

-26.37

-25 80

-2522

The worst values of 'Vz and '¥1 when algonthm 2 applied every 1. 4, 8 or 16 samphng mstants over sequence 3

Table 711

Parameter

Every 4T

Every ST

'~'•

-28 54

-2513

w,

-2929

-2529

The worn values of \jl2 and \jl3 when algonthm S evaluates or prechcts the roots every 4 or 8 samphng mstants over sequence 1

Table 712

Parameter

Every 4T

Every ST

'1'1

-3138

-31 38

w,

-3171

-31 43

Table'713

The worst values of 'l'z

and 'tf1 when algonthm S evaluates or predicts the roots every 4 or 8 samplmg

mstants over sequence 2.

Parameter

Every 4T

Every ST

'~'•

-3011

-30 11

w,

-26 89

-2637

Tablc714

The worst values of 'Vz and

'VJ when algonthm S evaluates or predtcts the roots every 4 or 8 samplmg

mstants over sequence 3

Table 715

.

Operation

Every4T

Every ST

Addttton& Subtractron

2611

1331

Multtphcauon

2148

1095

DtVlSIOO

48

24

Average number of anthmeuc operauons mvolved when algonlhm 5 evaluates or predtcts the roots every 4 or 8 samplmg mstants over sequence 1.

227

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - ----- --

Ope rattan

Every 4T

Every ST

Addmon& Subtracuon

2860

1442

Mu!uphcauon

2442

1185

D.vtston

48

24

Average number of anthmeuc operauons mvolved when algonthm S evaluates or predtcts the roor.s every 4 or 8 samplmg mstants over sequence 2

Table 7 16

Ope ratlon

Every4T

Every ST

Ad